Taking from the infinite.
I was unable to take any calculus classes due to being in special ed (because of stupid) so I never learned very much about infinites so bare with me. If someone took a single drop of water of finite size from an infinite ocean would it actually be taking from the ocean? Would the ocean replace that exact drop immediately upon it being taken or would it simply never matter? I assume there could be no butterfly effect and nothing could really be changed by it? Is the drop a free gift?
Comments (251)
I think not! Some nerve. :gasp:
No butterfly effect. Just the opposite. I wouldn't say the ocean is infinite. It just seems that way.
I don't think one can go much further on this topic. But, as usual, I could be wrong.
And a cup of water! Go figure.
if the ocean was truly infinite there would be no way to take a cup of water from it
cups would not even exist, nor would any space outside of the ocean to move it to
An island on an infinite ocean.
your trying to describe a finite infinity
which is a contradiction
nothing is created or destroyed only changed
When I took my wallet out of my pocket, I created a nothing in my pocket.
air is in your pocket
so your wallet and the air just traded places
therefore nothing was created or destroyed only traded
How wrong I was, and how deep the posts that followed . . .
Quoting MikeListeral
illusions are temporary
reality is eternal
I disagree, an ocean is an expanse of salt water essentially, which would contain of all the facets of an ocean on earth, given that the terms of the ocean was not specified in the question, it would be an ocean similar in all parts to the ocean on earth, given that that is the defining feature of the ocean. Thus, cups could be found in shipwrecks, and perhaps in garbage patches.
then an infinite ocean is impossible and would never exist
Infinite is a quality, not a quantity. It has to do with our limits of perception in relation to the ocean.
I agree that an infinite ocean would be impossible
The infinite ocean would have an infinite amount of drops for you to take. You would still be able to take drops, thete would just be an infinite number to take.
I think the drop being removed wouldn't matter as to the size of the ocean but that doesnt mean there would be no “butterfly effect”, it just means whatever that butterfly effect happens to be it will not effect the infinity of the ocean.
Quoting Bradaction
The equivalent of putting your hands over your ears and shouting "Nah, nah, nah..." in the hope that the problem will go away.
Sure, infinite oceans are impossible.
But if we had one, and we took a drop from it, the ocean would still be infinite.
That's what infinity does.
Yep, the drop is a free gift.
Where would one put the drop of water? Who would take it. Of course, is an infinitely expanding ocean fundamentally the same as an infinite ocean?
In the cup.
Quoting Bradaction
Me. Or @TiredThinker. Either will do.
Quoting Bradaction
Yes, except that it is expanding.
What threads like this show is that folk have odd notions of infinity.
Tell that to a mathematician.
To be fair infinity is an odd notion to start with. I learned that one infinity can be bigger than another infinity and that was even stranger. Its a niche concept imo, but yes, odd.
Fun, but.
Then we get things like:
Quoting Possibility
Which is just wrong.
Depends on the mathematician. For me, usually infinity is unboundedness, hence a quality of sorts.
Indeed, a term being hijacked and slaved to a specific position or argument. Par for the course round these parts.
'infinity' as a name occurs as sometimes, such as points in the extended real system. But that is different from the sense 'is infinite'. And such points are not required themselves to be infinite sets, though in some treatments they are.
'to infinity' in such statements as regards limits is not a separable term but abides only in the structure of a larger notation.
It's right. The quantities are the particular cardinalities. No cardinality itself is 'infinity'. Rather, each infinite cardinality has the property of being infinite.
One infinite set may be larger than another infinite set.
Infinities - all of them - are cardinalities.
Like odd numbers.
Thanks, I appreciate the correction.
What is your mathematical definition of 'infinites'?
Unlike the odd numbers, there is no set of all infinite sets nor of all infintie cardinals. So what are the infinities?
We could define "x is an infinity iff x is infinite'. But then 'is an infinity' is a predicate and doesn't stand for a quantity.
A cardinality is a quantity. But 'is infinite' is an adjective.
It's not a merely pedantic distinction. Ignoring the distinction causes real confusions about sets and set theory. Ignoring the distinction is typical of cranks (not you) who know nothing about set theory but try to refute it with incoherent arguments that conflate 'is infinite' as if 'infinity' is a noun.
Actually all the infinities are ordinals. Even the cardinals are ordinals these days, though they didn't use to be. That was all explained in the post I wrote that you were kind enough to thank me for, while announcing that you weren't going to read it. It's true that most people have heard about the transfinite cardinals and not the ordinals, but FWIW, ordinals are logically prior to cardinals, in the modern formulation.
Quoting Banno
LOL. But thank you for saying thanks!
Can you put a time for it when this change happened?
why waste your time trying to solve non existent problems
can god make a square circle? its not a limit of god. its just wordplay
infinite ocean is just wordplay. it doesn't eixst. therefore you dont need to solve any questions about it
1923. Found this in John von Neumann and Hilbert's School of Foundations of Mathematics (pdf link)
"The definition of ordinals and cardinals was given by von Neumann in the paper Zur
Einführung der trans niten Zahlen (1923)"
They also mention that von Neumann was still cleaning up his definition in 1928, since definitions by transfinite recursion were on shaky ground in that era.
The modern definition is the von Neumann cardinal assignment. Von Neumann defined a cardinal as the least ordinal having that cardinality.
Prior to that, cardinals were the equivalence class of all sets having that cardinality. The problem was that this was a proper class and not a set, so you couldn't manipulate cardinals using the rules of set theory. Von Neumann's definition defines each cardinal as a particular set, which is more convenient.
Quoting MikeListeral
The unit circle in the taxicab metric is a square. There's a picture of a square circle on that page. Better to use "married bachelor," because in fact there are square circles!
Quoting MikeListeral
"Good sense about trivialities is better than nonsense about things that matter." -- Quote on a math professor's door that I saw once.
That's pretty good. I like it.
Perhaps then the question can be asked, are vertical infinity, horizontal infinity and infinity, all potentially different terms that could be given to different types of infinity?
'is infinte' can be qualified any way you can come up with a definition of your qualifier.
is countably infinite
is uncountaby infinite
is infinte in correspondence with the y axis
is infinite in correspondence with the x axis
Etc.
Should we then refer to these terms as different types of infinite?
I wouldn't. I would say they are different predicates of the form: x is infinite & Rx.
Isn't this circular? Doesn't "least" already imply cardinality, such that cardinality is already inherent within the ordinals, to allow the designation of a least ordinal? Then the claim that ordinals are logically prior to cardinals would actually be false, because more and less is already assumed within "ordinal".
First let's begin with finite sets:
A = {w, o, r, k}
A has 4 elements
Take away the element w: A - {w} = B
and
I'm left with {o, r, k}
Element w was taken away. Effects:
1. The set B doesn't have the element w
2. The cardinality of B (3) is less than the cardinality of A (4)
The missing element in B produces a corresponding decrease in the cardinality of B (from 4 elements in A to 3 elements in B).
Let's now look at infinite sets:
N = {1, 2, 3,...}
Take away 1 as in, N - {1} = M = {2, 3, 4,...}
Effects:
3. M is missing the element 1
4. The cardinality of N = The cardinality of M = Infinity
The missing element in M (which is 1) fails to produce an effect on the cardinality of M.
See the difference?
A. Oceans aren't defined in terms of unions of droplets.
This means that atomically constructive definitions of oceans in terms of merging droplets together is irrelevant in terms of the logical characterisation of an ocean that assumes no physics. To mathematically define an ocean is to write it down instantaneously without constraining it's size.
B. Oceans are potentially infinite in terms of their number of droplets, but are not actually infinite.
This means that
1) An ocean is Dedekind-finite; there does not exist a constructable bijection between any number of droplets extracted from the ocean and a proper subset of those droplets.
2) An ocean is not specifiable a priori as a finite object in the sense that there is no a priori specifiable upper-bound on the number of droplets that can be extracted from it. In other words, an ocean, apriori, isn't equivalent to any a finite subset of droplets extracted from it. In mathematical parlance, oceans are therefore Kuratowski-infinite, like an infinite-loop in a computer program that isn't a priori equivalent to any finite number of loop iterations.
Together, 1 and 2 necessitate the rejection of the Axiom of Countable Choice, since that axiom forces all non-finite sets to be dedekind infinite.
Oceans are streams in a type-theoretical sense, which are lazily-evaluated lists
Ocean (0) = Ocean (no droplets so far extracted)
Ocean ( n) := [ droplet (n+1), Ocean (n+1) ] (n+1 droplets so far extracted)
Therefore we can say Ocean(0) > Ocean(1) > Ocean (2) .... without assigning a definite quantity to Ocean (0) and its predecessors, and without assuming that Ocean(i) is evaluated for all i, in the sense that only when we draw a droplet from ocean (i) does ocean (i) expand into [droplet(i+1), ocean (i +1) ].
And when the ocean eventually runs dry, our non-standard mathematical specification that is consciously aware of an a priori/ a posteriori distinction in mathematical meaning, isn't contradicted by reality, unlike in the case of classical set theory that in appealing to AC equivocates the a priori with the a posteriori.
OK, I will...
...Yep, I stand by my statement.
No, although it's slightly tricky. We are distinguishing between two sets having the same cardinality -- meaning that there is a bijection between them -- and assigning them a cardinal -- a specific mathematical object that can represent their cardinality.
Quoting Metaphysician Undercover
No, "least" is in terms of ordinality, not cardinality.
Quoting Metaphysician Undercover
Well it is, I agree with that. Take for example the ordinals [math]\omega[/math] and [math]\omega + 1[/math]. They have the same cardinality, as they can be represented by two distinct orderings of the same set, as I endeavored to explain to you in the other thread. But they are different ordinals, with [math]\omega < \omega + 1[/math].
So yes, cardinality is already inherent within the ordinals. Each ordinal has a cardinality. In the old days, before von Neumann, we identified a cardinal number with the class of all sets having that cardinality. After von Neumann, we identified a cardinal with the least ordinal of all the ordinals having that cardinality. The benefit is that the latter definition makes a cardinal into a particular set; whereas the former definition is a class (extension of a predicate) but not a set.
Quoting Metaphysician Undercover
Any nonempty collection of ordinals always has a least member, by the definition and construction of ordinals.
Quoting Metaphysician Undercover
Not at all. Not "more or less," but "prior in the order," if you prefer more accurate verbiage.
You insist on conflating order with quantity, and that's an elementary conceptual error. In an order relation x < y, it means that x precedes y in the order. x is not "smaller than" y in a quantitative sense. I can't do anything about your refusal to recognize the distinction between quantity and order.
Mine?
I just go to Wolfram: an unbounded quantity that is greater than every real number.
That way I can share the blame.
No prob, I regretted not adding a smiley to my earlier post. I didn't expect anyone to read all that, but the tl;dr is that ordinals are an important class of transfinite numbers even though fewer people have heard of them than have heard of cardinals. Smileys to make up for previous omission. :-) :-) :-) :-) :-)
Yes, there are points of infinity on the extended real line. So if by 'infinities' we mean such points and others in different number systems and such, fine.
But you said they are cardinalities. It is not required that such points have infinite cardinality, though in some treatments they might. Cardinality is a different subject.
That Wolfram article is poorly conceived.
It defines 'infinity' in the sense of points such as on the real line.
But then it mentions infinity with regard to infinite sets. Notice that 'infinite' is the word there, not 'infinity'.
'infinity' is a noun. So it is a name for a certain object, such as the point of positive infinity on the extended real line. Note that that point does not itself have to be an infinite set, even if in some treatments it may be.
'infinite' is an adjective not a noun. It is not name of a certain object and so it is not the name of a certain cardinality. Rather, it is property of certain sets and a property of certain cardinalities.
The Wolfram article sets up confusion by glibly conflating 'infinity' with 'infinite'.
As I've said previously in this thread and elsewhere, and now again:
'infinity' - a noun - does not refer to a cardinality. It couldn't even do that, since there are many infinite cardinalities. But 'infinity' may refer to things like a point on the extended real line.
What refers to cardinality is 'infinite' - an adjective. The predicate 'is infinite' applies to sets, as a set either is or is not infinite.
Wolfram uses "real", which I suppose is better than "cardinal" or "Ordinal" a
s a definition.
But it's authoritative that it's a quantity, not a quality, contra . So I'm worried by Quoting jgill.
I'm not wanting to contradict our resident mathematician.
I think the expectation is that folk will look at the related articles for more detail.
Or alternately, I'll have to disavow it's authority, which I am loath to do.
The [math]\pm \infty[/math] of the extended real numbers are not the same as the transfinite ordinals and cardinals. The extended reals are the standard reals with two meaningless symbols [math]\pm \infty[/math] adjoined, and given certain formal properties such as [math]a + \infty = \infty[/math] and so forth, entirely for the purpose of being able to say things like, "as x goes to infinity" rather than, "as x gets arbitrarily large." The extended reals are a notational convenience in calculus and integration theory. They should not be confused with the transfinite ordinals and cardinals. I didn't look at the Wolfram article but if they contributed to this confusion, then bad Wolfram!
ps -- Ok I looked at the article. First they start out by talking about infinity as one of the points adjoined to the real line to make the extended reals. Then they casually conflate this to Cantor's work.
Bad Wolfram. Bad article.
What does 'it' refer to there?
That's a terrible excuse. One shouldn't initiate further study by first publishing a dictionary entry that conflates important concepts.
I agree with the basics in your post.
One technical point though:
Yes, in many (probably most or even just about all) writings, the points of infinity are just arbitrary points, and they are not specified to be any particular mathematical objects. But in some treatments, the points are specified to be certain objects, so that the set of reals with extensions is a definite set.
It is not required that the extension points have infinite cardinality.
So what would your replacement be?
I did not think this was an appropriate context in which to mention the two-point compactification of the real line. Do you? You must have driven your teachers crazy. That's ok, I did too.
I'd cobble together some of my remarks here with some other stuff. Whatever I did, I would make clear that 'infinity' and 'infinite' are not be be conflated, and explain that as I have here.
Doesn't have to be that. Could be just to choose any two mathematical objects that are not real numbers for +inf and -inf. For example, +inf = w ('w' for omega, standing for the set of natural numbers). Then the system is a certain specific mathematical object. Not a major point in context of this thread; but it is a technicality that should not be deined.
Was it denied? Or simply omitted according to the common-sense principle of responding to a question at the level at which it was asked?
I refer you to Rudin, Principles of Mathematical Analysis (pdf link), for decades the standard undergrad text for math major real analysis. On page 11 of the linked edition he says that the extended reals are the reals with two symbols adjoined.
So you are wrong on the pedagogy AND wrong on the math. Not for the first time.
Quoting TonesInDeepFreeze
Maybe I should call your parents.
I did. In fifth grade, the teacher showed a wall map of the acquisitions of U.S. territory. The map omitted the Gadsen Purchase and included it in the Mexican Cession. I said aloud in class that the map is wrong. She said it's not. I explained that it omits the Gadsen Purchase and that land was not obtained by the Mexican Cession. She said to be quiet. I told her that I would be quiet when she told the class that they should understand that the map is wrong. My parents were called. They told me not to argue with the teacher.
I did not intend to imply that you personally denied it. But rather that it should not be denied.
You said what the extended reals are. I noted a qualification.
I did not say that you were personally amiss for not including that qualification, nor that you were not reasonable to deem it as too much detail for your purposes.
Merely, I added stated the qualification, and said that it should not be denied, while not meaning to imply that you personally denied it.
It's in your obfuscatory and unnecessarily argumentative mind that anyone denied it. You just made that up. Nobody denied anything. And I just gave you a link to Rudin, the number one classic real analysis text, that defines the extended reals exactly as I did, as the reals with two symbols adjoined having certain formal properties. You are wrong on the pedagogy AND wrong on the facts. My friend, if it's in Rudin, it's right. End of story.
Your pickiness with everything I write annoys me. Especially because half the time you're actually wrong on the facts. I got bored of arguing with you in the other thread and I've achieved the same state of blissful detachment here.
It may be that your pickiness annoys me because I have the same turn of mind, and we are always annoyed by those qualities in others that remind us of ourselves. That said, I've had enough for one day.
I didn't say that anyone denied it. I said it shouldn't be denied. And if that was not clear, I followed up in reply to say I did not intend to imply that it was denied.
And there was no obfuscation.
Quoting fishfry
Yes, I said in my original post:
Quoting TonesInDeepFreeze
I introduced my point to grant that. Then I said that we can also handle it another way. It's not unreasonable for me to say that.
Quoting fishfry
(1) I didn't make a claim about pedagogy. (2) t's not pedagogically inappropriate to add, essentially a footnote in this case, a certain technical qualification. (3) Posts don't need to restrict themselves to what is pedagogically best anyway.
EDIT: And I'm not wrong on the substance.
I just laughed, man. I think we're two of a kind. Peace.
It is not picky for me to say that you start your post with exaggeration: (1) There is a vast amount of what you write that I don't respond to, let alone with disagreement, correction, or qualification. (2) My points are not mere pickiness. That is only your own characterization.
And whether something annoys you, you blow it way out of proportion, and often seemingly taking it to be improperly motivated against you.
Quoting fishfry
If half the time I'm wrong, then the other half I'm right or at least neutral. Moreover, I have not been wrong half the time or anywhere remotely close.
I started my post by saying that I basically agree with your post. Then I said I'm adding only a technical qualification. And I even said that most writers don't use the method I am mentioning but that others do and that it can be done. That's pretty damn mild.
Then after one of your posts, I said that my point should not be denied. I didn't say that you denied it. My point was that no one should deny it. Then when you asked whether it had been denied and that you didn't mention it so as not to complicate things. Then I stated explicitly that I did not mean to imply that you denied it. And I will even add here that I don't blame you for thinking that I did mean to imply that you denied it. But after I've said it now about three times, you may correctly infer that I mean what I say when I say that I did not mean to imply that you denied it.
It is literally true that I did not say anyone denied it. And I haven't said that I blame you if you think I meant to imply that you denied it. And I don't blame you if you think that I meant to imply that you denied it. And above (though cross-posted) I say again that I did not mean to imply that you denied it.
Quoting fishfry
Yes, you are the rational, logical, reasonable, accommodating, conciliatory, factually correct, patient and pedagogically sagacious one. You are the Gallant of The Philosophy Forum. And I am the irrational, illogical, unreasonable, unaccommodating, non-conciliatory, factually incorrect, impatient, and pedagogically unwise one. I am the Goofus. And you're the better looking one too. The roles of hero and villain have been clearly scripted. Now for casting.
Quoting fishfry
and Love, man.
Well then it's incorrect to say that ordinality is logically prior to cardinality. If there is already cardinality inherent within ordinality then the closest you can get is to say that they are logically codependent. But if order is based in quantity, then cardinality is logically prior.
Quoting fishfry
"Least", lesser, and more, are all quantitative terms. So as long as you are using "least" to define your order, it is actually you who is conflating quantity with order. If you want a distinct order, which is not quantitative, you need something like "before and after", or "first and second". But first and second is a completely different conception from less and more, and would not be described by "least".
If you want to emphasize a difference between quantity and order you need to quit using quantitative words like "least", when you are talking about order. However, I should remind you, that "least" is the term you used for Von Neumann's definition. If Von Neumann used the quantitative word "least", in his definition, then I think it is just a faulty interpretation of yours, which makes you insist on distancing quantity from order. In the reality of mathematical practise, order is defined by cardinality, not by anything like "first and second". So cardinality is held to be logically prior, regardless of what you claim.
Ah Thank you!
There aren't any infinite oceans.
You're absolutely right. It would be incorrect to say that, because it's not true. I was pretty sure that I HADN'T said that, and I went back to page 2 of this thread and found what I actually said:
Quoting fishfry
As you see, I said that ordinals are logically prior to cardinals. That's because cardinal numbers ARE particular ordinals. That's correct. That's what I said.
Now I don't think your misquote of me was deliberately disingenuous. Rather, I think you don't have the mathematical sophistication to follow this conversation at all. Because in my previous post to you, I already explained the distinction between cardinality, which is an equivalence relation based on bijection; and cardinal numbers, which are particular ordinals. I see that went right over your head, leading to you inaccurately quote me based on your ignorance about what I already explained to you previously.
And frankly I'm not going to get into it with you about this stuff. Go read my long article on the transfinite ordinals, or read the relevant Wiki page, or read a book on set theory. I can't argue with you about established, universally-accepted math. Unless you want to tell me what you think you know that John von Neumann didn't.
Quoting Metaphysician Undercover
You lack the understanding to even know what you're saying. Again: Cardinality is inherent. How you define a cardinal number isn't. That you don't understand the distinction shows that you need to do a little homework on your own before you can credibly engage on this topic. There is no philosophical point involved. Two sets may have the same cardinality, without there being any notion of ordinal at all. But cardinal numbers are defined as particular ordinals. Cardinal numbers are subtly different than cardinality. I explained this to you previously, you either didn't read it or didn't understand it (not an exclusive or) and went ahead and deliberately misquoted me. It's tedious.
Quoting Metaphysician Undercover
Yeah yeah. I can't help you out. You should make an honest attempt to learn this material. I have made mighty efforts to explain basic order theory to you. You don't want to hear it. I am under no obligation to get into yet another conversation about this. I'm going to take my cue from John von Neumann here.
In other words, you agree that it's incorrect to say that ordinals are logically prior to cardinals. That is, unless you are just trying to hide a vicious circle by saying that a cardinal number is defined by its ordinality, and ordinality is defined by cardinality. But in the case of a vicious circle of two logically codependent things, it is still incorrect to say that one is logically prior to the other. So despite my lack of understanding of your "bijective equivalence", it is still you who is mistaken.
No, I said ordinals are logically prior to cardinals, in the modern von Neumann interpretation. I explained this several times. It's not right for you to hijack yet another thread by pointlessly trolling me like this.
Quoting Metaphysician Undercover
Two sets are bijectively equivalent if there is a bijection between them. In that case we say they have the same cardinality. We can do that without defining a cardinal number. That's the point. The concept of cardinality can be defined even without defining what a cardinal number is.
But you say it's "my" bijective equivalence as if this is some personal theory I'm promoting on this site. On the contrary, it's established math. You reject it. I can't talk you out of that.
Yep.
Well, this is all fascinating on a mathematical level :yawn:, but the question in the OP was:
Quoting TiredThinker
1. You can’t actually take from an infinite ocean, because there isn’t actually an infinite ocean to begin with. An ocean might appear infinite, but given that you are not an aspect of the ocean (and that you can remove a drop) renders any ocean you can speak of potentially finite (ie. at the very least it ends where you begin).
2. An ocean is not a static, measurable object but an event - an ongoing process of evaporation and precipitation - and so is indeterminately quantifiable in terms of finite drops of water. Once you remove a single drop and recount, the quantity of finite drops of water in said ocean will have changed anyway, so there’s no way you could tell if you’d made any difference at all, even if you could immediately count all the drops. This is quantum physics.
3. It could matter to the organisms living inside the drop that was removed, though. I couldn’t really say.
There. You can go back to talking about infinities now...
There are probably modern complex variables people who have more sophisticated ideas about complex infinity. Directed infinity is one such notion, in which the line has a particular slope (or the point at infinity has an argument). And then there are the various geometries and ideal points, manifolds, etc.
So, there's quite a bit more about infinities or infinite than what is found in set theory. As I have mentioned, when I play with dynamical systems, infinite means unbounded in whatever context it appears and I rarely speak of the point at infinity :cool:
I do not presume to speak for fishfry, but I would like to state some points, and add some points, in my own words too.
fishfry:
On matters of logic and mathematics, any divergence I take from you or qualification I mention is not intended as a criticism of you personally. (I am not suggesting that you have claimed or not claimed that I have intended my remarks on logic and mathematics to be personal criticisms of you.) My personal criticisms have only been about the dialogue itself.
I am not suggesting that you would be remiss if you didn't adopt my formulations and definitions.
I am not suggesting that my sometimes more detailed formulations supplant your sometimes less formal explanations (though in some instances I think your formulations are not correct).
When I simply add detail or additional commentary, I am not suggesting that your original remarks are thereby incorrect.
I am not suggesting that you would be remiss by not reading or not replying to anything I write (except rebuttals that are defenses against your incorrect criticisms of my claims).
When I state something that you have already stated, I am not suggesting that you had stated to the contrary nor that you hadn't already stated it.
I am not suggesting that you would be remiss not to the include the details I include.
I am not suggesting that threads such this one require the detail that I include.
I am not claiming that my formulations are pedagogically superior, as my intentions are not purely, or even primarily, pedagogical.
I am not suggesting that my comments supplant yours.
I am not suggesting that posting should be expected to keep a level of precision as we may expect in professional publication.
If I misconstrue a poster, then it is unintentional. I do not intend to cause a strawman. If I have misconstrued a person, then they can let me know. But they would be incorrect if they claimed my error was intentional or that I had intentionally set up a strawman. (And I am not suggesting that you have or have not claimed that I have set up a strawman in previous threads.)
I try to write mostly at face value. But in any communication it is often not clear whether a person meant to imply more than they literally said or not. If I have seemed to imply something that I did not literally write, then anyone can ask me whether I meant to imply it or not.
/
I am posting because I like talking about mathematical logic.
I like expressing the concepts and explaining them.
Sometimes my explanations are not understandable for people who are not familiar with mathematical logic, and in that case, I still enjoy having explanations and formulations available possibly for posters to revisit or even I enjoy just the fact that my remarks are on the record.
I enjoy making formulations that are as rigorous as feasible in the confines of a thread.
Posting sharpens my knowledge of the subject and improves my skill in composing formulations.
When I post corrections to other posters, I find some small satisfaction in seeing that the correction is available to be read.
I hope that some readers might benefit from my posts.
I believe many of my formulations do provide insight and rigor and at least examples showing that rigorous formulations of certain notions exist.
Sometimes I enjoy the interaction with posters.
I enjoy reading some posters.
I benefit from any true corrections or suggestions presented to my own posts.
And with cranks, I find entertainment and satisfaction in providing counter to them.
/
Quoting fishfry
That is correct and it is important. It points to the fact that the set theoretic treatment of ordinals and cardinals is rigorous as it proceeds only step-by-step through the theorems and definitions.
Quoting fishfry
That is correct and it is important. For every S, there is the equivalence class of all sets that have a bijection with S. But that equivalence class is a proper set. So for a rigorous set theory, without proper classes, another definition of the cardinality operator needed to be devised. The numeration theorem ("for every S there is an ordinal T such that there is a bijection between S and T") is a theorem of ZFC, and it allows us to define card(S) = the unique ordinal k such that k has a bijection with S and such that no ordinal that is a member of k has a bijection with S.
Quoting fishfry
That is correct and it is important. A set theoretic operator (such as 'the cardinality of') can be defined only by first showing that for every S, there exists a unique T such that T has a [fill in a certain property here]. For example, with the operator 'card' (meaning 'the cardinality of') we first prove:
For every S, there is a unique T such that T is an ordinal and T has a bijection with S and no member of T has a bijection with S.
card(S) = the unique T such that T is an ordinal and T has a bijection with S and no member of T has a bijection with S.
Quoting fishfry
I see the main point there, but the formulation is not clear to me. I suggest this sequence:
df: k is ord-less-than j <-> k e j
When context is clear, we just say "k is less than j" or "k < j" or "k e j".
df: k is ord-least in S <-> (k e S & ~Ej(j e S & j ek))
When context is clear, we just say "k is least in S".
df: k is the least ordinal such that P <-> (k is an ordinal & Pk & ~Ej(Pj & j ek))
df: the cardinality of S = the unique T such that T is the least ordinal having a bijection with S
So the cardinality of S is card(S).
df: S is a cardinal <-> (S is an ordinal and there is no ordinal T less than S such that there is a bijection between S and T)
Quoting fishfry
That is correct and important. No clear understanding of ordinals and cardinals can be had without it.
Quoting fishfry
That is correct and important especially in context of being presented in this thread with misconceptions about this. The less than relation on ordinals is simply the membership relation. That is, membership ALONE is the basis for the less than relation on ordinals. But a general quantitative relation for sets is formulated with cardinals that are based both on bijection and instantiated to specific sets with regard to the ordinal less than relation.
Quoting fishfry
I see what you intend, but to be precise, a relation is a set of tuples. But cardinality is not a set of tuples. The equivalence relation is among the cardinalities but is not the cardinalities themselves.
Quoting fishfry
I agree with the intent of that and I think perhaps some authors say things like that in a sense that does not require ordinals, but I find it not quite right.
There are two different notions:
(1) 'the cardinality of S' to mean the least ordinal k such that there is a bijection between S and k.
and
(2) 'S and T have the same cardinality' to mean there is a bijection between S and T
In (1) 'cardinality of' is a 1-place operation.
In (2) 'same cardinality' is a 2-place predicate.
That's okay, except:
(2) S and T have the same cardinality iff there is a bijection between S and T.
In that sense, we don't need to rely on ordinals.
(3) S and T have the same cardinality iff card(S) = card(T).
In that sense, we do rely on ordinals.
Which of (2) or (3) is the definition of 'same cardinality'? We wouldn't know unless the author told us, and whether the definition relies on ordinals would depend on what s/he told us the definition is.
And (3) is better than (2) in the sense that (3) uses 'cardinality' compositionally from (1) while (2) takes 'cardinality' noncompositionally. And (3) better fits the usage such as: "What is the cardinality of S? It's the least ordinal k such that there is a bijection between S and k. Okay, so S and T have the same cardinality iff the cardinality of S is the cardinality of T."
So I would just define:
S and T have the same cardinality iff card(S) = card(T).
Then take (2) as a theorem not a definition.
And if we want to leave out ordinals, then just say "There is a bijection between S and T".
/
To clean up two misconceptions that have been expressed in responses to fishfry:
Incorrect: The notion of ordinals presupposes the notion of cardinality.
A definition of 'is an ordinal' does not refer to 'cardinal' nor 'cardinality', and it doesn't even refer to 'bijection'. That is a plain fact that can be verified by looking at any textbook on set theory.
A definition of 'the order type of' does not refer to 'cardinality nor 'cardinality'. That is a plain fact that can be verified by looking at any textbook on set theory.. (The term 'ordinality' has been used. I am not familiar with it. Perhaps 'the ordinality of' means 'the order type of'?)
Incorrect: We should not use 'least' if we don't mean quantity.
It is typical of cranks unfamiliar with mathematical practice to think that the special mathematical senses of words most conform to their own sense of the words or even to everyday non-mathematical senses. The formal theories don't even have natural language words in them. Rather, they are purely symbolic. Natural language words are used conversationally and in writing so that we can more easily communicate and see concepts in our mind's eye. The words themselves are often suggestive of our intuitions and our conceptual motivations, but proofs in the formal theory cannot appeal to what the words suggest or connote. And for any word such as 'least' if a crank simply could not stomach using that word in the mathematical sense, then, if we were fabulously indulgent of the crank, we could say, "Fine, we'll say 'schmleast' instead. 'schmardinality' instead'. 'ploompty ket' instead of 'empty set' ... It would not affect the mathematics, as the structural relations among the words would remain, and the formal symbolism too.
LOL I think you made your point. It's all good. Maybe you can straighten out @Metaphysician Undercover :-)
I don't know what your point is there.
No, I think you misinterpret this. I say it's "your" bijective equivalence, because you are the one proposing it, not I. So "yours" is in relation to "mine", and anyone else who supports your proposition (even if you characterize it as "established math") is irrelevant. If you wish to support your proposition with an appeal to authority that's your prerogative. In philosophy, the fact that something is "established" is not adequate as justification.
Quoting fishfry
This is what I do not understand. Tell me if this is correct. Through your bijection, you can determine cardinality. But are you saying that you do this without using cardinal numbers? What is cardinality without any cardinal numbers?
What I think is that you misunderstand what "logically prior" means. Here's an example. We define "human being" with reference to "mammal", and we define "mammal" with reference to "animal". Accordingly, "animal" is a condition which is required for "mammal" and is therefore logically prior. Also, "mammal" is logically prior to "animal". You can see that as we move to the broader and broader categories the terms are vaguer and less well defined, as would happen if we define "animal" with "alive", and "alive" with "being". In general, the less well defined is logically prior.
Now let me see if I understand the relation between what is meant by "cardinality" and "cardinal number". Tell me if this is wrong. An ordinal number necessarily has a cardinality, so cardinality is logically prior to ordinal numbers. And to create a cardinal number requires a bijection with ordinals, so ordinals are logically prior to cardinal numbers.
Where I have a problem is with the cardinality which is logically prior to the ordinal numbers. It cannot have numerical existence, because it is prior to ordinal numbers. Can you explain to me what type of existence this cardinality has, which has no numerical existence, yet is a logical constitutive of an ordinal number.
There is no mention of 'cardinal' or 'cardinality' in the definiens.
df: the cardinality of S = the least ordinal k such that there is a bijection between S and k.
There is no mention of 'cardinal' or 'cardinality' in the definiens.
/
I don't use the term 'logically prior', but in context it probably would be fully explicated by induction on terms. Basically that a term T is prior to term Y iff the definiens in the definition of T does not depend on Y but the definiens in the definition of Y does depend on T. What requires induction on terms is the notion of 'depends'.
In that sense 'ordinal' is prior to 'cardinal'
When you do, you will be enlightened. :-)
I'm not proposing it, I'm reporting it from Cantor's work in the 1870's. You wouldn't call it "my" theory of relativity, or "my" theory of evolution, just because I happened to invoke those well-established scientific ideas in a conversation. Minor semantic point though so let's move on.
Quoting Metaphysician Undercover
So if I name-drop the theory of relativity or the theory of evolution I have to provide evidence? And if I don't I'm merely appealing to authority? Not an auspicious start to a post that actually did get better, so never mind this digression as to who gets credit for the idea of cardinal equivalence, and what is my burden of proof for simply mentioning that Cantor thought of it 150 years ago.
Quoting Metaphysician Undercover
Through a bijection we can determine cardinal equivalence. If two sets X and Y have a bijection between them -- something that can be objectively determined -- we say they are cardinally equivalent. We still don't know what a cardinal number is. We only know that X and Y are cardinally equivalent.
Quoting Metaphysician Undercover
There's cardinal equivalence without cardinal numbers. If there's a bijection between X and Y, then X and Y are cardinally equivalent. But we still haven't said what a cardinal is.
It's a bit like saying that the score in a baseball game is tied -- without saying what the score is. Maybe that helps. Or in the classic example of bijective equivalence, I can put a glove on my hand and determine that the number of fingers on the glove is the same as the number of fingers on my hand, simply by matching up the glove-fingers to the hand-fingers bijectively. But that doesn't tell me whether the number of fingers is 4, 5, or 12. Only that the number of fingers is the same on the glove and on my hand, by virtue of matching the fingers up bijectively.
So via establishing a bijection between the glove-fingers and the hand-fingers, I can say that the number of fingers is cardinally equivalent between the glove and my hand. But I still can't assign a particular cardinal number to it.
Quoting Metaphysician Undercover
If one thing is defined in terms of some other thing, the latter is logically prior. As is the case with cardinal numbers, which are defined as particular ordinal numbers.
Quoting Metaphysician Undercover
I'd agree that given some ordinal number, it's cardinally equivalent to some other sets. It doesn't "have a cardinality" yet because we haven't defined that. We've only established that a given ordinal is cardinally equivalent to some other set.
Quoting Metaphysician Undercover
So we have this notion of cardinal equivalence. We want to define a cardinal number. In the old days we said that a cardinal number was the entire class of all sets cardinally equivalent to a given one. In the modern formulation, we say that the cardinal number of a set is the least ordinal cardinally equivalent to some given set.
Note per your earlier objection that by "least" I mean the [math]\in[/math] relation, which well-orders any collection of ordinals. If you prefer "precedes everything else" instead of "least," just read it that way.
[I should put in a parenthesis here, we are not actually defining cardinal numbers, but rather only the Alephs. In the presence of the axiom of choice, they're all the same. All the cardinals are Alephs. In the absence of choice, there are cardinalities that aren't Alephs. This is not anything we should care about today].
Quoting Metaphysician Undercover
No. Cardinal equivalence is logically prior to ordinals in the sense that every ordinal is cardinally equivalent to some other sets. At the very least, every ordinal is cardinally equivalent to itself.
When you use the word "cardinality" you are halfway between cardinal numbers and cardinal equivalence, so you confuse the issue. Better to say that cardinal equivalence is logically prior to ordinals; and that (in the modern formulation) ordinals are logically prior to cardinals.
Quoting Metaphysician Undercover
Well we should banish the word cardinality, because it's vague as to whether you mean cardinal equivalence or cardinal number. What kind of existence does a bijection between two sets have? Well a bijection is a particular kind of function, and functions have mathematical existence. In fact if X and Y are sets, and f is a function between them, then f is a set too. So whatever kind of mathematical existence sets have, that's the kind a bijection has.
Some philosophers would say that functions have a higher "type" than sets, but we're not doing type theory, and in set theory everything is a set, at the same level. But if your mathematical ontology puts functions into a different level of existence than sets, then whatever level functions live in, that's what a bijection is. A bijection is just a kind of function.
Well, at least thank you for not saying 'thank you'.
I said LOL because I was amused/charmed by your lengthy pre-apologies and disclaimers before providing your commentary on my technical points. I thought you went overboard but that you probably felt that you needed to go to those lengths to placate me. Which at that moment I found amusing. If that makes sense. But that's what was behind the LOL.
I didn't get why you chose that clause in particular. I see now - it was just the nominated example.
(Disclaimer yes, apology and placation no.)
Oh right, good point. Yes a "nominated example," great phrase. A synecdoche, as it were, a part that stands in for the whole, like "all hands on deck."
Quoting TonesInDeepFreeze
I took it as very effective placation, so over-the-top I had to laugh.
So a notation X (primitive or defined) is prior to another notation Y (defined) iff the definition of Y depends on X. (1) So this is relative to the sequence of definitions; different treatments of a theory, even with all the same set of defined notations, may have different sequences of definitions, (2) We need a definition of 'depends on'.
The notations that are defined for set theory are function symbols, predicate symbols, and variable binding operators such as the abstraction operator and the definite description operator. I'm leaving out the variable binding operators for now, because even giving a rigorous definition of the variable binding operators is complicated and requires double induction. So by 'notation' here I mean just function symbols and predicate symbols.
I haven't yet come up with a definition of 'depends on'. Intuitively it's that Y depends on all the notations that appear in the definiens for Y, and the notations that appear in the notations in the definens for Y, and finitely backwards until we reach the primitives. So it's inductive. And for a notation there's a tree, not a sequence, back to the primitives. For example in set theory:
n is even <-> (n is a natural number & Ek(k is a natural number & n = 2*k))
So 'even' depends on 'natural number', and '2' and '*'. And each of those depend on previously defined notations, and downwards in a tree to the primitives '=' and 'e'.
But 'even' (or any other notation) could also have been defined in the primitive language alone, without using any intermediary notations. This may make a non-syntactical notion of 'prior' problematic. For example, neither 'ordinal' nor 'cardinal' is non-syntactically prior since both could be defined themselves using only the primitive 'e'. Of course, in practice, the definition of 'cardinal' has 'ordinal' in the definiens. But that is not necessary, as 'cardinal' could also be defined from just '=' and 'e'. Of course such a definition of 'cardinal' would be a massive formula and impractical for people to work with. But 'practical' is not formalized, and what we are investigating is formal syntax. In principle, even if it would not be practical, 'cardinal' can be defined form 'e' alone.
Anyway, for 'prior' I need
Tree(Y) = [fill in formal definition of the tree of notations that branches up to the definition of Y]
X is prior to Y iff X is a node in Tree(Y)
df: K is a cardinal <-> (K is an ordinal & Aj(j e K -> there is no bijection between j and K))
["K is a cardinal iff (K is an ordinal and there is no bijection between K and an ordinal less than K"]
df: card(x) = the least ordinal j such that there is a bijection between x and j
["the cardinality of x is the least ordinal that has a bijection with x"]
theorem: Ax card(x) is a cardinal
["every cardinality is a cardinal"]
theorem: Aj(j is an ordinal -> EK card(j) = K)
["every ordinal has a cardinality"]
If we adopt a particular systematic and explicit sequence of definitions, and eschew locutions that don't "interock" with one another, then we leave fewer openings for being strawmanned by cranks.
Quoting fishfry
That's really good.
Some people claim that classifications must obey certain essentialities in order to be correct. For example (I'm not trying to state the more complicated actual zoological taxonomy) a claim that only this classification is correct:
animal (mammal (canine, feline, ...), reptile (snake, lizard ...) ,,,)
But that notion is not viable. For example:
(1) passenger vehicle (Ford (Fusion, Mustang, ...), Honda (Accord, Civic ...), ...)
(2) passenger vehicle (sedan (Ford, Honda, ...) van (Ford, Honda, ...), ,,,)
Both (1) and (2) may be pertinent depending on our purpose.
And we may be wary of the essentialist mistake in mathematics.
Continuum Hypothesis?
That article is good because it's hard to find layman's terms explanations of forcing and the proposed axioms.
But a couple of points:
"Cantor realized that [the set of natural numbers is 1-1 with the set of odd numbers]".
He "realized" it? It was known for at least 300 years. And probably a lot longer.
"In addition to the continuum hypothesis, most other questions about infinite sets turn out to be independent of ZFC as well."
Questions aren't independent. Sentences are. There are only countably many sentences. So the cardinality of the set of theorems is equal to the cardinality of the set of independent sentences.
/
"Kennedy, for one, thinks we may soon return to that “prelapsarian world.” “Hilbert, when he gave his speech, said human dignity depends upon us being able to decide things in mathematics in a yes-or-no fashion,”"
I hadn't read the speech. That is really interesting about human dignity. I understand why we would seek a decision procedure, but why would Hilbert think our dignity depends on it? So incompleteness would lead Hilbert to abandon hope of human dignity? This is really interesting.
But there are infinite more natural numbers, just as with the reals. Is the point that there are far more infinities of reals than infinity of naturals vs the odd? I can imagine putting any two infinities one to one if you start with one number, then two, and onward
Please. If you have a rigorous definition of "infinite more" different from set theoretic "greater cardinality" then fine, state your definition, and your claim could be right relative to that definition. Meanwhile the card(N) = card(set of odds) is a theorem of set theory.
Quoting Gregory
I can't parse that.
Quoting Gregory
It is a theorem of set theory that no set is 1-1 with its power set. It's a theory of set theory that N is not 1-1 with R.
If you have an alternative theory, then state your axioms. What you merely imagine is not mathematics.
It seems to me the natural numbers are a type of power set to the odd. If I imagine (not a bad word) any infinity as a ruler going off east into forever, I can make this ruler and a second by taking the first number of a countable set and the first number of a uncountable set and send them off to infinity like you do with naturals and odds.Why isn't it the same thing? The rulers would fit side by side because they are all within infinity
Impressionistic descriptions are fine for stoking creativity in mathematics and sometimes for making certain mathematical concepts intuitive. But they are not mathematical demonstrations.
If you know why we can do:
Odd numbers: 1 3 5 6....
Natural numbers: 1 2 3 4...
but can't do:
Countable: 1 2 3 4...
Uncountable: 1 2 3 4...
then say it.
I really want to know
By definition, there is no bijection between a countable set and an uncountable set.
By theorem, there is no bijection between N and R. The proof has been given thousands of times in textbooks, articles, and on the Internet (it's even outlined in the article linked to above). You really are not familiar with the proof?
Quoting Gregory
Then open an Internet search engine and type 'proof uncountability reals'.
No because I've asked people many times and they bring up the diagonal thing, although this just shows there are infinity more uncountable than countable and yes, however there are infinity many natural than odd. But you can biject with one and not the other? I'm not a jerk, just want some way I can understand what they are saying. It seems to me infinity is always just infinity at the end
You are terribly confused. You asked me to prove there is not bijection between a countable set and an uncountable set. And I told you where to find the proof. But you say you already know about it. So there was no point in asking me.
But you say there are infinitely many more odd numbers than natural numbers. So YOU prove that. Not mind pictures, but mathematical proof.
If you really do want to understand, then get a book on set theory and start reading it from page 1. I could type proofs and explanations for you all day, but if you don't have the background for it, then it's a waste. Any proof I give you will depend on proofs and definitions previous to that proof, on and on backwards until we reach the axioms. So that is utterly impractical in a thread. The reasonable and enlightened way is to get a book and read it from page 1, from the axioms through the proofs.
It seems to me you can state the reason you can move the odd numbers in line with the naturals but can't move the countable in line with the uncountable in a pretty simple way say that I can go and find more about this
Have you seen the simple and beautiful proof of Cantor's theorem? It shows that there is no possible surjection from a set to its powerset.
Here's the proof. Let [math]X[/math] be a set, [math]\mathscr P(X)[/math] its powerset, the set of all subsets of [math]X[/math]. Let [math]f: X \to \mathscr P(X)[/math] be a function, and by way of starting a proof by contradiction, suppose [math]f[/math] is a surjection.
Since [math]f[/math] is a function that inputs an element of [math]X[/math] and outputs some subset of [math]X[/math], for any given element [math]x \in X[/math] it may or may not be the case that [math]x \in f(x)[/math]. Let [math]S[/math] be the subset of [math]X[/math] defined by [math]S = \{x \in X : x \notin f(x)\}[/math].
Now [math]S[/math] is a subset of [math]X[/math]; and since by assumption [math]f[/math] is a surjection, there must be some element [math]s \in X[/math] such that [math]f(s) = S[/math].
Now we ask the question: Is [math]s \in S[/math]? Well if it is, by definition it isn't; and if it isn't, by definition it is. Therefore the assumption that there is such an [math]s[/math]; that is, that [math]f[/math] is a surjection; leads to a contradiction.
Therefore there is no surjection from any set to its powerset.
To take an example, consider the set {a,b,c}. Its powerset is {ø, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}. It's perfectly clear that there's no surjection from a 3-element set to an 8-element set; and this principle holds even in the infinite case.
Cantor's theorem immediately gives us an endless hierarchy of infinities [math]\mathbb N, \mathscr P(\mathbb N), \mathscr P( \mathscr P(\mathbb N)), \dots[/math] In terms of the simplicity of the argument versus the profundity of the result, I don't think there's anything comparable in all of mathematics.
Wow that's simply put. Thanks
You are welcome! So glad that worked for you. This argument is much simpler and more natural than the diagonal argument, it should be better known.
We don't need to suppose toward contradiction that there is a surjection.
Let f:X -> PX
Let S = {y e X | ~y e f(y)}
S e PX
S e ran(f) -> EyeX f(y) = S
Let f(y) = S
y e f(y) <-> ~ y e f(y)
So f is not a surjection
OK Tones, explain to me then what "least" means in "the mathematical sense", if it is not a quantitative term. It can't be "purely symbolic" in the context we are discussing. For example, when fishfry stated von Neumann's definition of a cardinal as "the least ordinal having that cardinality", through what criteria would you determine "least", if not through reference to quantity?
Quoting TonesInDeepFreeze
Here's another example. Look at your use of "less than". How is one ordinal "less than" another, without reference to quantity?
Quoting fishfry
Yes, in the context of the example we are discussing, I would. Unless you were quoting it word for word from another author, or explicitly attributing it to someone else, I would refer to it as your theory. I believe that is to be expected. Far too often, Einstein's theory, and Darwin's theory are misrepresented,. So instead of claiming that you are offering me 'Cantor's theory', it's much better that you acknowledge that you are offering me your own interpretation of 'Cantor's theory', which may have come through numerous secondary sources, unless you are providing me with quotes and references to the actual work.
Quoting fishfry
OK, so let's start with this then. In general we cannot determine that a game is tied without knowing the score. However, if we have some way of determining that the runs are equal, without counting them, and comparing, we might do that. Suppose one team scores first, then the other, and the scoring alternates back and forth, we'd know that every time the second team scores, the score would be tied, without counting any runs. Agree? Is this acceptable to you, as a representation of what you're saying?
Quoting fishfry
Here's where the problem is. You already said that there is a cardinality which inheres within ordinals. This means that cardinality is a property of all ordinals, it is an essential, and therefore defining feature of ordinals. So we have a sense of "cardinality" which is logically prior to ordinals, as inherent to all ordinals, and we also have a sense of "cardinal" number which is specific to a particular type of ordinal.
Quoting fishfry
Don't you see how this is becoming nonsensical? What you are saying is that it has a cardinality, because it is cardinally equivalent to other sets, but since we haven't determined its cardinality, it doesn't have a cardinal number. In essence, you are saying that it both has a cardinality, because it is cardinally equivalent, and it doesn't have a cardinality because it's cardinality hasn't been determined, or assigned a number.
Let's look at the baseball analogy. We know that the score is tied, through the equivalence, so we know that there is a score to the game. We cannot say that because we haven't determined the score there is no score. Likewise, for any object, we cannot say that it has no weight, or no length, or none of any other measurement, just because no one has measured it. What sense does it make to say that it has no cardinal number just because we haven't determined it?
Quoting fishfry
Actually, this explains nothing to me. "Precedes" is a relative term. So you need to qualify it, in relation to something. "Precedes" in what manner?
Quoting fishfry
Yes, this demonstrates very well the problem I described above. Because the set has a "cardinal equivalence, it also necessarily has a cardinality, and a corresponding mathematical object which you call a cardinal number. Why do you think that you need to determine that object, the cardinal number, before that object exists as the object which it is assumed to be, the cardinal number?
I was largely thinking in terms of the movie, "Dr. Strange" in which the Ancient One keeps her youth by stealing energy from the dark dimension which I was led to believe is an infinite world as perhaps our universe maybe as well. So basically 2 or more infinite universes at least in the fiction exists. The Ancient One had an ethical dilemma over her theft from the dark dimension even though in theory nothing is lost, but I wasn't sure. Many physicists believe there could be many worlds split into more worlds whenever a paradox needs to be resolved so Schroder's Cat can actually be both alive and dead, but in different worlds.
Yes you are correct, it's cleaner to not use proof by contradiction. Thanks for the clarification.
In general, a set can be infinite yet not include "everything." For example there are infinitely many even numbers, but they don't include the odd numbers.
In the case of an infinite ocean we have to work a little harder to get a good visualization. Maybe the world is like a 3D chessboard, with an infinite ocean on one level and an infinite plane of land on the next. So we can be sitting on land and reach down to take a cup of water from the infinite ocean below. Now we have a cup of water; but since the ocean is liquid, it immediately fills up the space where we removed the water, and there's still an infinite ocean.
What do you think? Visualization-wise, I mean?
The essential idea though is that you can always take a finite amount from an infinite set, and the set is still infinite. But it doesn't necessarily have to be all of what it was before. I believe @TheMadFool gave this example earlier, where we can start with the infinite set 1, 2, 3, 4, ..., then remove 1 to leave 2, 3, 4, ... What's left is still infinite, yet it's missing 1. That can happen too. Infinity is funny that way.
ps Here's another idea. The world is a flat, infinite plane. Like "flat earth" theory except instead of a great wall of ice around the edge, it just goes on forever. The entire world is a vast ocean, but there are infinitely many finite-sized islands spread throughout. So there's an island here and another one there, infinitely many in all, but they're all separated by water. So if you're on land you can always dip your cup in the infinite ocean.
If you drop a pebble in the ocean it will ripple forever. And if you drop lots of pebbles there will be lots of ripples, all the time.
We would continue to prove that the uncountability of Pw implies the uncountability of R:
It suffices to prove that the interval [0 1] is uncountable.
We have the theorem Ax Px 1-1 with 2^x.
So P^w 1-1 with 2^w.
And we prove 2^w injects in [0 1]
For finite sets,
1. K = {a, b, c}, L = {a}
2. Set difference: K - L = {b, c}
3. K - L =/= K
Where n(A) is the number of elements in set A,
n(K) = 3
n(L) = 1
4. Arithmetic difference: n(K) - n(L) = 3 - 1 = 2
5. n(K) - n(L) =/= n(K)
---------------------
For infinite sets,
6. N = {1, 2, 3,...} O = {1}
7. Set Difference: N - O = {2, 3, 4,...}
8. N - O =/= N
Where n(A) is the number of elements in set A,
n(N) = Infinity
n(O) = 1
9. Arithmetic difference: n(N) - n(O) = Infinity - 1 = Infinity
10. n(N) - n(O) = n(N)
I answered this in my most recent post to you. Given two ordinals, it's always the case that one is an element of the other or vice versa. So for ordinals [math]\alpha[/math] and [math]\beta[/math], we define [math]\alpha < \beta[/math] if it happens to be the case that [math]\alpha \in \beta[/math]. This is perfectly well-defined and unambiguous, especially in the case of the von Neumann ordinals which are constructed exactly so that this works out. [math]\emptyset \in \{\emptyset \} \in \{{\emptyset, \{\emptyset\}\}} \in \dots[/math], and that's exactly how ordinal "less than" is defined. It's also the case that the von Neumann ordinals are defined in such a way that [math]\in[/math] is transitive; if [math]\alpha \in \beta[/math] and [math]\beta \in \gamma[/math] then [math]\alpha \in \gamma[/math]. This is NOT true of sets in general, but it IS true for ordinals, and that's what makes the construction work.
Quoting Metaphysician Undercover
Via the [math]\in[/math], which is a primitive in set theory and is always true between any two ordinals in one direction or the other, by construction of the ordinals.
Quoting Metaphysician Undercover
The Wikipedia articlea on ordinals, cardinals, Cantor, etc. are perfectly satisfactory in this regard.
Quoting Metaphysician Undercover
Rather than try to save that example, I'll just repeat the hand/glove example. By putting on a glove, I can determine whether my hand-fingers are in bijective correspondence with the glove-fingers, without knowing the actual cardinal number.
c
Here's where the problem is. You already said that there is a cardinality which inheres within ordinals.
I went to great pains to note last time that your use of the word "cardinality" is ambiguous and causing you to be confused. You should either say cardinal equivalence or cardinal number, to clearly disambiguate these two distinct but related notions. Every ordinal number is inherently cardinally equivalent to many sets. If nothing else, every ordinal is cardinally equivalent to itself, so the point is made.
But we still don't know which cardinal number that is.
And again, when you say "cardinality," you obfuscate the distinction between these two concepts.
Quoting Metaphysician Undercover to many other sets, including itself. But when we clarify this terminology, your sophistic point evaporates.
Quoting Metaphysician Undercover
No, as I'm pointing out to you. It's true that every ordinal is cardinally equivalent to itself, but that tells us nothing. You're trying to make a point based on obfuscating the distinction between cardinal numbers, on the one hand, and cardinal equivalence, on the other.
Quoting Metaphysician Undercover
No no no no no. I hope I've explained this.
Quoting Metaphysician Undercover
Other way 'round. A cardinal number is defined as a particular ordinal, namely the least ordinal (in the sense of set membership) cardinally equivalent to a given set.
I hope you can see that by carefully using the phrases, "cardinal equivalence" and "cardinal number" properly, all confusion goes away. You are deliberately introducing confusion by using the word "cardinality" ambiguously.
Quoting Metaphysician Undercover
No, it's very carefully thought out by 150 years worth of mathematicians including von Neumann, widely agreed to be "the smartest man in the world" by his contemporaries. It's your insistence that everyone else is wrong about things that you aren't willing to put in the work to understand that's nonsensical.
Quoting Metaphysician Undercover
No. I am repeatedly telling you to stop using that word, because you are using it to confuse yourself. It's true that every ordinal is cardinally equivalent to various other sets including itself; and it's true that a cardinal number is defined as a particular ordinal.
Your entire argument is based on obfuscating the word cardinality. You should stop, because I can only explain this to you so many times without losing patience. I already explained this to you repeatedly in my previous post.
Quoting Metaphysician Undercover
Right. I can live with that. I know I have the same number of fingers as my glove, but I don't know how many fingers that is.
Quoting Metaphysician Undercover
No. I'm telling you to stop using the word cardinality until you understand what's being talked about.
Quoting Metaphysician Undercover
This is you just continuing to confuse yourself over the word cardinality. If you'll just carefully say cardinal equivalence when you mean that, and cardinal number when you mean that, we might make progress.
Quoting Metaphysician Undercover
Because cardinal numbers are a [i]defined term[/url]. Given a set, we have to build a sophisticated technical apparatus in order to define what we mean by its cardinal number.
But I could take a step back from all this. My remark about what's logically prior to what is true, but it's not that important in the scheme of things. It's more important for you to make an effort to understand what ordinal numbers are, because they're important. So if all you care about it to be right about the logically prior business, that's the wrong thing to care about. It's not an important matter.
Quoting Metaphysician Undercover
Given two ordinal numbers, it's always the case that one is an element of the other, as sets. We define [math]<[/math] as [math]\in[/math]. If you prefer you can always think of the [math]\in[/math] whenever I say that one ordinal is "less than" another, or that some ordinal is the "least" with such and so property.
Quoting Metaphysician Undercover
Cardinal equivalence is a relation between two sets. It's not something a set can have by itself. So it is not true that "a set has a cardinal equivalence" in isolation. That makes no sense. I can say, "My brother and I have the same parents," but it makes no sense to say, "I have the same parent." As what?? Having the same parent is a relation between two things. It doesn't stand alone. Likewise two sets may be cardinally equivalent to each other or not. But a set doesn't have cardinal equivalence by itself, that makes no sense.
Quoting Metaphysician Undercover
Please stop using that word till you have a better grasp on the material. Your constant misuse of it is only causing you confusion.
Quoting Metaphysician Undercover
Yes, it has that, after we've built up a whole bunch of theory to define what that is.
Quoting Metaphysician Undercover
Because after Cantor defined cardinal equivalence, the question came up among philosophers, "What actually is a cardinal?" At first they did the obvious thing, they said a cardinal was the entire class of all the sets that are cardinally equivalent to a given set. That is a perfectly satisfactory definition, but it suffers from the flaw that such a class is not a set. Von Neumann figured out how to define cardinal numbers as particular sets, so that they could be manipulated using the rules of set theory.
It's just a matter of wanting to define what a cardinal number is, in formal terms. So that when we have two sets that are cardinally equivalent to each other, we have a collection of canonical sets such that every set is cardinally equivalent to exactly one of them.
I see where you're going with this. Given a set, it has a cardinal number, which -- after we know what this means -- is its "cardinality." You want to claim that the set's cardinality is an inherent property. But no, actually it's a defined attribute. First we define a class of objects called the cardinal numbers; then every set is cardinally equivalent to exactly one of them. But before we defined what cardinal numbers were, we couldn't say that a set has a cardinal number. I suppose this is a subtle point, one I'll have to think about.
Here's an example. Whenever I have a party I like to put everyone in separate rooms according to their approximate height. I have ten rooms and I arrange the people so that there's more or less an equal number of people in each room. So at the party, each person is a "room 1 person" or a "room 2 person" and so forth.
But when you got up that morning, before you came to my party, you weren't a room 3 person or whatever. The assignment is made after you show up, according to a scheme I made up. Your room-ness is not an inherent part of you.
Likewise, given a set we can assign it a cardinal number. But it's far from clear that this is an inherent property of a set. Rather, we set up the scheme of defining cardinal numbers so that given a set, we can figure out which cardinal it's assigned to. It's an after-the-fact defined assignment, not an inherent quality.
Another example, a bunch of people show up for a work detail. I assign some to dig ditches, some to cut down trees, some to supervise, some to do this, some to do that.
Before I made the work assignments, the jobs were not inherent properties of the people. Rather, I assigned those jobs after the fact. Just as I can take a set and assign it a cardinal number. But of course in the case of a set's cardinal number, that's a more subtle question. Did the set "inherently" have a cardinal number before I assigned it?
Good point, if that is your point. I'll give this some thought.
But notions that have been given mathematical definition include:
ordinal
well ordering
ordinal less-than
least ordinal in a set
least ordinal with a formula defined property
bijection
equinumerous
isomorphism
order type
injection
cardinal
cardinality
cardinal less-than
Definitions adhere to forms that ensure ensure eliminability (formulas with defined terms can be set back to formulas without the defined terms) and non-creativity (formulas that weren't already provable aren't made provable with the introduction of defined terms). By adherence to the forms for definitions, the definitions are never circular.
And new definitions can be provided - anyone is free to introduce a new term and define it.
Set theory is not properly critiqued by acting as if some undefined terms controls results in set theory.
theorem: For every S, there is a unique T such that T is an ordinal & S and T are equinumerous & no member of T is equinumerous with S
df: card(S) = the unique T such that T is an ordinal & S and T are equinumerous & no member of T is equinumerous with S.
We say card(S) is the cardinality of S.
df: k is ord-less-than j <-> k e j
When context is clear, we just say "k is less than j" or "k < j" or "k e j".
df: k is ord-least in S <-> (k e S & ~Ej(j e S & j ek))
When context is clear, we just say "k is least in S".
df: k is the least ordinal such that P <-> (k is an ordinal & Pk & ~Ej(Pj & j ek))
df: S is a cardinal <-> (S is an ordinal & there is no ordinal T less than S such S and T are equinumerous)
theorem: S is a cardinal <-> Ex S = card(x)
/
There is no circularity there.
If one has a definition of 'inherent' then they can add it.
OK, this makes more sense than what you told me in the other post, that one "precedes" the other. You are explaining that one is a part of the other, and the one that is the part is the lesser..
Quoting fishfry
I assume that an ordinal is a type of set then. It consists of identifiable elements, or parts, some ordinals being subsets of others. My question now is, why would people refer to it as a "number"? Say for instance that "4" is used to signify an ordinal. What it signifies is a collection of elements, some lesser than others. By what principle is this group of elements united to be held as an object, a number? Do you know what I mean? A set has a definition, and it is by the defining terms that the sameness of the things in the set are classed together as "one", and this constitutes the unity of the set. In the case of the "ordinals", as a set, what defines the set, describing the sameness of the elements, allowing them to be classed together as a set?
Quoting fishfry
The issue, which you are not acknowledging is that "cardinal" has a completely different meaning, with ontologically significant ramifications, in your use of "cardinally equivalent" and "cardinal number".
Let me explain with reference to your (I hope this is acceptable use of "your") hand/glove analogy. Let's take the hand and the glove as separate objects. Do you agree that there is an amount, or quantity, of fingers which each has, regardless of whether they have been counted? The claim that there is a quantity which each has, is attested by, or justified by, the fact that they are what you call "cardinally equivalent". So "cardinal" here, in the sense of "cardinally equivalent" refers to a quantity or amount which has not necessarily been determined. Suppose now, we determine the amount of fingers that the hand has, by applying a count. and we now have a "cardinal number" which represents the amount of fingers on each, the glove and the hand. In this sense "cardinal" refers to the amount, or quantity which has been determined by the process of counting.
Quoting fishfry
Do you agree with this characterization then? An ordinal is a type of set, and a cardinal is a type of ordinal. Logical priority is given to "set". So do you agree that a cardinal number is not an object, but a collection of objects, as a set? Or, do you have a defining principle whereby the collection itself can be named as an object, allowing that these sets can be understood as objects, called numbers?
Quoting fishfry
But this is an inaccurate representation. What you are saying, in the case of "cardinal numbers", is not "that. I know I have the same number of fingers as my glove, but I don't know how many fingers that is", but that there is no "number" which corresponds with the amount of fingers in my glove, until it has been counted and judged. You can say, I know I have the same "amount" of fingers as my glove, but you cannot use "number" here, because you are insisting that the number which represents how many fingers there are, is only create by the count.
Quoting fishfry
But you already said a set can be cardinally equivalent with itself. "If nothing else, every ordinal is cardinally equivalent to itself, so the point is made."
Quoting fishfry
Yes this exemplifies the ontological problem I referred to. Let's say "cardinality" is a definable attribute. Can we say that there is a corresponding amount, or quantity, which the thing (set) has, regardless of whether its cardinality has been determined? What can we call this, the quantity of elements which a thing (set) has, regardless of whether that quantity has been judged as a number, if not its "cardinality"?
Quoting fishfry
I see this as a very dangerously insecure, and uncertain approach, epistemically. See, your "scheme" is completely arbitrary. You may decide whatever property you please, as the principle for classification, and the "correctness" of your classification is a product simply of your judgement. In other words, however you group the people, is automatically the correct grouping.. The only reason why I am not a 3 person prior to going to the party is that your classification system has not been determined yet. If your system has been determined, then my position is already determined by my relationship to that system without the need for your judgement. It is your judgement which must be forced, by the principles of the system, to ensure a true classification. My correct positioning cannot be consequent on your judgement, because if you make a mistake and place me in the wrong room, according to your system, you need to be able to acknowledge this. and this is not the case if my positioning is solely dependent on your judgement.
If you go the other way, as you are doing, then the position is determined by your subjective judgement alone, not by the true relation between the system of principles and the object to be judged. So if you make a mistake, and put me in the wrong room, because your measurement was wrong, I have no means to argue against you, because it is your judgement which puts me in group 3, not the relation between your system and me.
Df. If x and y are ordinals, then x precedes y (x is less than y) iff x is an element of y.
Th. If y is an ordinal, then for all x, if x is a member of y, then x is a subset of y.
Th. If y is an ordinal, then y = {x | x is an ordinal & xey}. I.e., every ordinal is the set of its preceding ordinals.
/
Ordinals are called 'ordinal numbers'. But that is not needed for the actual formal results in set theory. We could just as easily always say 'ordinal' instead of 'ordinal number'. However, 'ordinal number' does reflect that there are operations on ordinals - ordinal arithmetic, ordinal multiplication, and ordinal exponentiation.
'is a number' is not a predicate of set theory. Saying 'ordinal number' rather than 'ordinal' does not confer any special property of being a "number"; it's just suggestive phrasing and does not add any inferential force.
/
There is no set of all the ordinals.
/
'is an ordinal' was defined, at least a few times, already in this thread.
/
fishfry is using 'cardinally equivalent' to mean 'equinumerous' (also 'equipollent'). the term 'cardinally equivalent' does not depend on having previously defined 'is a cardinal' nor 'the cardinality of'.
Again:
df. x and y are cardinally equivalent iff there is a bijection between x and y
[neither 'ordinal' nor 'cardinal' are mentioned in that definition]
Probably a more common terminology is:
df. x and y are equinumerous iff there is a bijection between x and y
df. x is a cardinal iff (x is an ordinal & ~Ey(yex & x and y are equinumerous))
df. card(y) = the cardinal x such that x and y are equinumerous
th. Ax(x is a natural number -> x is an ordinal)
th. {x | x is a natural number} = {x | x is an ordinal & x is finite
/
IMPORTANT SUGGESTION:
One should always understand that English terminologies such as 'ordinal number', 'equinumerous', etc. are only nicknames for actual formal symbols in set theory. The nicknames are suggestive of certain intuitions and motivations, but the nicknames do not confer any deductive effect. The only things that definitions that provide only abbreviation of actual formulas written solely in the primitive language.
It is a common error for people who are not familiar with formal theories to think that we can make deductions from the themes suggested in the nicknames; for example, saying "Well, if they're ordinal numbers then they must have all the properties of other numbers such as natural numbers, rational numbers, et. al}. That's wrong.
So, again, it is incorrect to think that 'cardinally equivalent' has some kind of "ontological" connection with 'is a cardinal' or 'the cardinality of'.
Again, three separate things (and the suggestiveness of the English word 'cardinal' in each of them does not confer any deductive force):
x and y are equinumerous [fishfry is calling that 'x and y are cardinally equivalent']
x is a cardinal
x is the cardinality of y [or said by, 'x = card(y)']
and:
th. x is a cardinal iff Ey x = card(y) [and 'x is a cardinality' just means 'x is a cardinal']
/
Every set is an object. Every cardinal is a set. Every cardinal is an object.
But 'object' is not a term in the language of set theory. It's just a word we use to talk about the things (objects) that are ranged over by the quantifier in the primitive language.
When talking about Z set theories, one can use 'set' and 'object' interchangeably since Z set theories don't "talk about" anything other than sets.
Is that a thing? Ok.
But @TonesInDeepFreeze doesn't appear to be eschewing proof by contradiction, instead merely proving (still by contradiction) a stronger denial of surjectivity than mere failure of surjectivity. Hence his supposing ("toward contradiction") that the denial is false amounts to supposing less than necessarily complete surjectivity, which would entail the whole power set being in the range of f, and amounts instead to supposing merely the presence of S in the range of f.
Thus showing, that the naturals can't be used to keep count of their own groupings/combinations/sub-sets if any of them (naturals) are needed to index (I mean count, map to) groupings they aren't in. Because that would create the set S, which would need to but couldn't without contradiction be in the range of f.
Hope I've got that right. And if I have, then the suggestion to prove only the more specific failure of surjectivity has helped me, at least.
Then again, Tones hasn't exactly signposted the supposition, that S is in the range of f (the supposition being, I thought, in order to show that it leads to contradiction), and he doesn't even explicitly state it. It just (as line 4 perhaps alludes but doesn't actually say) follows from
Quoting TonesInDeepFreeze
which (I guess?) follows from
Quoting TonesInDeepFreeze
So, I don't know. Maybe they were undecided whether it was meant to be read as a proof by contradiction or not.
For some mathematicians its a stylistic preference.(I'm not sure, but I think maybe my version is Cantor's version.)*
Quoting bongo fury
The proofs prove the exact same result - nothing more nothing less, except for a stylistic choice. Both proofs are correct and intutionistically valid.
Quoting bongo fury
That's backwards.
EyeX f(y) = S
Let f(y) = S
That's existential instantiation.
EyeX f(y) = S is itself an RAA premise within the proof. In that way, firshfry's RAA is deferred in my proof to later.
I didn't write EyeX f(y) = S as a separate line, since I didn't belabor certain obvious steps; it's not a fully formal proof.
/
RAA and modus tollens are basically the same. RAA as a rule in natural deduction(or a derived rule from a Hilbert style system) while modus tollens does the job as an axiom in a Hilbert style system.
/
* Re Canor's diagonal proof, if I recall correctly, he doesn't start with a premise that there is a surjection and then derive a contradiction to infer there is not a surjection. Rather, he reasons about any arbitrary enumeration and shows that it is not a surjection. And I've read certain mathematicians say that they prefer not to set it up as an RAA. Same with the infinitude of the primes and other results. I don't know any philosophical reason for that; I take it as a stylistic choice.
That incorrectly makes it appear that I said, "Incorrect: We should not use 'least' if we don't mean quantity."
Sure, but yours begins (read as a proof by contradiction) by denying a more specific claim of failure of surjectivity: the claim that such sets as, in particular, S will fail to be in the range of f. Was my point. Obviously that more specific claim of failure implies the more general. But the denial of it is weaker than the denial of the more general. I thought this might be the correct way to interpret
Quoting TonesInDeepFreeze
(That you were saying we can suppose less.)
Quoting TonesInDeepFreeze
Sure. I was prepared to guess at quantifier introduction on the backwards journey, but the "I guess" probably sounded sarcastic. Without the sarcasm it probably doesn't improve much.
Quoting TonesInDeepFreeze
Yes. It's still a proof by contradiction, just not so upfront.
Quoting TonesInDeepFreeze
Sure. And it's still a proof by contradiction.
Quoting TonesInDeepFreeze
By RAA here I take it you mean the whole argument, while earlier it was a tag for the line that you
Quoting TonesInDeepFreeze
? Cool.
Quoting TonesInDeepFreeze
Another reason not to expect an important contrast in your reworking.
Quoting TonesInDeepFreeze
Quoting bongo fury
In the beginning, I didn't deny any claim claim whatsoever.
And, of course, I wouldn't even think of denying the claim that S is not in the range of f.
Quoting bongo fury
Yes, that is fair to say. Notice, I didn't say that RAA would be avoided in a (natural deduction style) proof. I only mentioned a particular opening RAA premise, viz. "There is a surjection" and that we don't need to adopt it as an RAA premise.
Quoting bongo fury
It has an RAA nested within. Of course.
Quoting bongo fury
As far as I can tell, the difference is merely stylistic. I said that in my previous post.
/
For reference:
Let f:X -> PX
Let S = {y e X | ~y e f(y)}
S e PX
S e ran(f) -> EyeX f(y) = S
Let f(y) = S
y e f(y) <-> ~ y e f(y)
So f is not a surjection
Except in a line properly signposted as RAA.
There's a double negative in what you're saying. RAA premise would not need to deny ~P. Rather, in this case, the premise is P.
Anyway, that's not the beginning of my proof.
It's perhaps a little cleaner in terms of exposition. Not a big deal either way.
Of course.
Quoting TonesInDeepFreeze
Exactly, if for some reason you want to label the RAA line "P" rather "~P".
In a line properly signposted as RAA, and in a discussion in which someone had bothered to say
Quoting TonesInDeepFreeze
it could make sense to display under that signpost (P or ~P depending on signposting preferences, or a form of words such as I chose so that the question didn't arise) the denial of what is to be shown in the argument. This denial will be the supposition toward a contradiction.
What is to be shown is that S can't, without contradiction, be in the range of f.
So the denial, the suitable RAA line, the supposition toward contradiction, if you or anyone did want to belabor the point, or understand the point about "not needing to suppose toward contradiction..." is indeed
"S is in the range of f",
and it might be interesting that this is taking the place of "f is surjective", in a proof by contradiction.
Quoting TonesInDeepFreeze
Agreed.
Quoting bongo fury
Quoting fishfry
How true.
Starting with an RAA premise provides a clear structure.
Not starting with RAA, but instead talking about an arbitrary function from S to PS, is a little more arch, and reveals the hammer blow more by surprise.
I wonder why Cantor didn't start with RAA.
In any case, the choice is not to be confused with eschewing intutionistically invalid RAA, since it is intuitionistically valid.
I don't know what you're saying.
Quoting bongo fury
You lost me.
No, you can't be bothered, and why should you.
My bad. Carry on.
I did bother. I read your post three times but couldn't figure it out.
(1) ~~P RAA premise ... contradiction ... infer ~P
and
(2) P RAA premise ... contradiction ... infer ~P.
(1) is not intuitionistically valid. (2) is intuitionistically valid.
I see that what I left tacit is not RAA, but just modus tollens:
Let f:X -> PX
Let S = {y e X | ~y e f(y)}
S e PX
S e ran(f) -> EyeX f(y) = S
Suppose EyeX f(y) = S [for -> introduction]
Let f(y) = S
y e f(y) <-> ~ y e f(y)
EyeX f(y) = S -> (y e f(y) <-> ~ y e f(y)) [-> introduction]
~EyeX f(y) = S [intuitionistically valid modus tollens]
So f is not onto PS
Another instance in which modus tollens does the same job as RAA.
That is what you said. You said the phrase, "We should not use 'least' if we don't mean quantity" is incorrect. I asked, if you don't mean some sort of quantity then what do you mean by "least". And fishfry gave me an answer to that.
Ok good.
Quoting Metaphysician Undercover
Yes. But that should be no surprise. In set theory everything is a set. There are no urelements in standard set theory. In math every single thing is a set. Numbers, groups, topological spaces, cardinals, ordinals, are all sets. Sets whose elements are sets whose elements are sets, drilling all the way down to the empty set. There is nothing but sets. Of course one need not found math on set theory, but in standard math, that's how it's done. Everything is a set.
Quoting Metaphysician Undercover
Yes. And those elements are sets. and those sets' elements are sets, all the way down to the empty set. Everything is a set. That's why they say math is based on set theory. Of course that's only historically contingent. Are numbers "really" sets? That's the question raised (and answered in the negative) by Paul Benacerraf in his famous essay, What Numbers Could Not Be.
Quoting Metaphysician Undercover
There's no general definition of number. Negative numbers didn't use to be regarded as numbers, nor did zero, irrational numbers, complex numbers. Quaternions are numbers these days, but William Rowan Hamilton got famous for discovering/inventing them in 1843. When Cantor introduced his cardinals and ordinals he got a lot of pushback from the mathematical community of the day, but in the end his point of view won out, and the transfinite ordinals and cardinals are numbers. What is a "number" is a matter of historical contingency.
Quoting Metaphysician Undercover
Ok. And lets be perfectly explicit. In the formalism, [math]0 = \emptyset, 1 = \{0\}, 2 = \{0, 1\}, 3 = \{0, 1, 2\}[/math] and [math]4 = \{0,1,2,3,\}[/math]. Of course this is only a formalism. As Benacerraf points out, the number 4 can't really "be" this set. Rather, it's just a particular representation. The number 4 is the abstract thingie pointed to by the representation. But we've had this conversation before.
Quoting Metaphysician Undercover
Groups of elements are united to be held as sets by the axioms of set theory. If X is a set and Y is a set then their union and intersection are sets, and so forth. You can find the axioms here.
Now what is an "object," I don't know, because object is a term of art in computer programming but not in math. And what's a number is, as I've pointed out, a matter of historical contingency. There are no principles other than Planck's great observation that scientific progress proceeds one funeral at a time. Meaning that the old guard die off and the young Turks grow up taking the new ideas for granted.
Quoting Metaphysician Undercover
Yes. You want to know what entitles [math]\omega^\omega[/math] to number-hood. Well it's the same thing that entitled [math]\sqrt 2, i[/math], and -47 to numberhood. Historical contingency. Someone said "Hey this weird thing is useful, I'll call it a number," and everyone else said, "You're crazy," and a generation or two later everyone called it a number. Simple as that. Human opinion over time, nothing deeper than that.
Quoting Metaphysician Undercover
No, not at all. First, the elements of a set need not be "the same" in any meaningful way. The only thing they have in common is that they're elements of a given set. The concept of "set" itself has no definition, as I've pointed out to you in the past. A particular set might have a specific definition; but even that sometimes fails, as in the nonconstructive sets given by various set-theoretic axioms. A set exists when the axioms say it does. To take a non-mathematical example, the set consisting of the number 5, the tuna sandwich I had for lunch, and the Mormon Tabernacle Choir may be taken together into a set consisting of three elements. Of course in math you can't have examples like that; the elements of sets have to be other sets. Unless you are working in a set theory that has urelements, which is a bit of a niche area.
Quoting Metaphysician Undercover
There is no set of ordinals, this is the famous Burali-Forti paradox.
What makes a particular set an ordinal is that it satisfies the textbook definition of an ordinal, namely a transitive set well-ordered by [math]\in[/math]. That technical definition needs to be unpacked, but that's the definition. If a set satisfies that definition, it's an ordinal.
Quoting Metaphysician Undercover
I've been pointing out to you the different meaning of cardinal equivalence and cardinal number for several posts now. I'm not sure why you claim I am not "acknowledging" that difference. I have been expending quite a few keystrokes to explain that distinction to you.
Quoting Metaphysician Undercover
I get the point you're making, it's an interesting philosophical point. If I define an odd number as a number that leaves a remainder of 1 when integer-divided by 2; and I then prove that 47 is an odd number; was 47 an odd number before I made the definition? It's a good question in mathematical philosophy. Not one we'll solve here today.
Quoting Metaphysician Undercover
Well not exactly. An ordinal is a type of set, yes. But a cardinal is not a "type" of ordinal at all. Rather, among all the ordinals cardinally equivalent to a given set, we take the least of them and designate that as the set's cardinal. So the cardinal-ness of an ordinal is not a property of an ordinal; rather, it's a name we give to an ordinal that has a particular property relative to a lot of other ordinals. Subtle point but important. It's a little like the captain of a football team. The captain is not a "type" of player; rather, the captain is a player that we have designated as the captain. The ordinal definition of cardinal is like that.
Quoting Metaphysician Undercover
In the sense that in the set-theoretic formalization of math, sets are fundamental. Ok. If that's what you're saying.
Quoting Metaphysician Undercover
I don't know what an object is (except in the context of everyday English, or computer programming; but not in math); so you'll have to tell me.
But a cardinal number is a set, yes. Everything is a set in set theory. Everything is a set.
Quoting Metaphysician Undercover
I have no idea what you mean by object. I only know about sets. The defining principles of what can be called sets are the axioms of set theory. What's called a number is a matter of human agreement, often hard-won over generations and always subject to revision.
Even the same object in different contexts is or isn't a number. A classic example is the number [math]i[/math], the imaginary unit. We call that a number. But we can model the complex numbers as a particular set of 2x2 matrices, and then we call them matrices and not complex numbers. The question of what's a number is a matter of human convention. There is no general definition of number.
Quoting Metaphysician Undercover
Well of course that's an interesting philosophical question, which we will not solve today. Was 47 an odd number before I defined what an odd number is? You want to say that somehow the cardinal numbers existed before we defined them. Fine, you're a Platonist today. Sometimes I am too. Other times, not so much. What of it? I agree it's a good question. Whether the cardinal numbers were "out there" waiting for von Neumann to come along and give them their definition; or whether he made them up out of his productive mind.
After all, in other posts you have cast personal doubt on the very existence of mathematical sets; and now you want to claim that cardinal numbers were already out there waiting for von Neumann to come along. You see you're at best a part-time Platonist yourself.
Quoting Metaphysician Undercover
If I put on my Platonist hat, I'll admit that the number 5 existed even before there were humans, before the first fish crawled onto land, before the earth formed, before the universe exploded into existence, if in fact it ever did any such thing.
And then I'll ask, well if the number 5 existed before the universe did, where did it exist? What else might live there? The Baby Jesus? The Flying Spaghetti Monster? Captain Ahab? Platonism is hard to defend once you start thinking about it.
I must say, though, that I am surprised to find you suddenly advocating for mathematical Platonism, after so many posts in which you have denied the existence of mathematical objects. Have you changed your mind without realizing it?
Quoting Metaphysician Undercover
Bit disingenuous there. The relation "has the same parents" is a binary relation, it inputs two people and outputs True or False. But they don't have to be distinct people. I have the same parents as myself.
Quoting Metaphysician Undercover
Which I fully acknowledge, and note that we are not going to solve it here. Were the transfinite cardinals out there waiting to be discovered by Cantor and then formally defined by von Neumann?
But I must note that I find it very strange to see you suddenly advocating for mathematical Platonism, after denying the existence of mathematical sets.
Quoting Metaphysician Undercover
Heck of I know. Did the number 5 exist before the Big Bang? Was it out there waiting for humanoids with five fingers to come along? Maybe you can answer me that first, before you ask me about the transfinite cardinals.
Quoting Metaphysician Undercover
I will agree that the fiveness of the fingers on my hand is not as arbitrary as my assignment of categories to people such as which room I put them in at a party, or who I designate as the captain of the team.
But @Meta, really, you are a mathematical Platonist? I had no idea.
Quoting Metaphysician Undercover
I agree with the points you're raising. I don't know if 5 existed before there were humans to invent math. I truly don't know if the transfinite cardinals were out there waiting to be discovered by Cantor, and formalized by von Neumann. After all, set theory is an exercise in formal logic. We write down axioms and prove things, but the axioms are not "true" in any meaningful sense. Perhaps we're back to the Frege-Hilbert controversy again.
My mistake.
I think you meant 'transitive set well ordered by ?'.
Thanks, I made the correction.
Could someone rightfully say that 0, 1, and points are not in any sense sets? Or is there more too that?
Sure. Euclid didn't have set theory but he talked about points. As far as the modern definition of numbers, there's Russell's type theory and its modern variants, there are category-theoret definitions, and so forth. I don't know much about any of these alternatives.
Benacerraf described it like this:
That is, the number 3 is not an object at all. Rather, it's a thing defined by its relation to other numbers. In his famous essay he kicked off the field of mathematical structuralism. @TonesInDeepFreeze already gave this link, I'll repeat it for reference.
https://plato.stanford.edu/entries/structuralism-mathematics/
Benacerraf's essay can be downloaded here. The quote above is found on page 70,
https://documents.pub/document/benacerraf-what-numbers-could-not-be.html
You have to click Download then it makes you wait 60 seconds. Other online links to the article either make you read it online or else don't let you read it at all. When I'm in charge, academic paywalls will be abolished. Taxpayers already paid for this research. I looked it up. Benacerraf worked at Princeton and Princeton takes Federal money.
In category theory there's a thing called a natural numbers object which is intended to capture the structural essence of natural numbers. I don't know much about this and the Wiki article isn't particularly enlightening.
Here's an article about the natural numbers type in modern type theory. It's also not very enlightening unless one is a specialist.
So the bottom line is that structuralists don't think that natural numbers "are" sets; or even that natural numbers are any particular thing at all. A natural number is whatever relates to other things the way natural numbers do.
ps -- Here's the Wiki article on mathematical structuralism.
https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_mathematics)
With a Hilbert style system, the axiom we use to derive modus tollens is given in the intuitionistically invalid form:
(~P -> ~Q) -> (Q -.> P)
and then we derive the intuitionistically valid form:
(P -> Q) -> (~Q -> ~P)
The reason is that we can't derive the intuitionistically invalid form from the intuitionistically valid form.
I didn't know about structuralism in math! That the number one is an idea, a true idea, seems to me to be the basis of all that follows though, kinda that unity before the plurality. But structuralism in all forms is a really interesting idea!
Or maybe we can think of vacuity and unity as the bases. Then we have 0 and 1, the binary. But in set theory, with the pairing operation, we can define '1' from '0'..
df: {x y} = z <-> Ak(k e z <-> (k = x v k= y))
dr: {x} = {x x}
df: x = 0 <-> Ay ~xey
df: 1 = {0}
df: n and m are natural numbers -> (n < m <-> n e m)
/
Set theory does provide the structural relations we expect. Even though the objects have "extra-structural" properties (e.g. that 0 is the empty set and 1 is the set {0}), the structural relations are captured (e.g. that 0 < 1).
I don't think there are any set theorists here. You're the only mathematician in the house. The rest of us, speaking for myself, are groupies and hangers-on at best. That said, can you say what "this" refers to? Cohen's invention of forcing in general? Or the particular recent result that's floating around the Internet about Martin's Maximum implying (*) or some such? The latter is some serious set-theoretic inside baseball.
I found the SEP article interesting. It breaks down all the various sub-genres of mathematical structuralism and talks a lot about whether category theory is an example of mathematical structuralism or not, and so forth. Lot of fancy philosophizing :-)
And they point out some of the drawbacks with structuralism. If the natural numbers are not any particular collection of objects, but rather are instances of some "structure," then what exactly is a structure?
The main point is that when we say that 0 is the empty set and 1 is the set containing 0, what we really mean is that these sets represent the natural numbers within set theory. What we don't mean is that these sets actually "are" the natural numbers. Leaving unanswered the question of what the natural numbers really are
My initial guess was that a set is something that contains and not something in its own right. So zero remains a nothingness of anything in that case. Very abstract ideas. Couldn't structure just be that which contains a process and thus, like sets which compose it, it is nothing in itself. This would certainly make mathematics a system of process and divorce it from the notion that anything rests and stays permanent within it
That's not the case in set theory.
If you replace numbers with sets, do you keep the same ontology? What of structuralism?
That would be good. I had heard the expression but had no idea what it was. The article came as a revelation to me. And here I thought the reals consisted of rationals and irrationals. :chin:
You and Tones are far more set theorists than me!
"Replace" might not be a good way of putting. A better way of putting it might be that sets "play the role" of numbers, or something like that.
And what ontology? Different philosophers and/or mathematicians have different conceptualizations of ontology for set theory.
Quoting Gregory
I addressed that:
Quoting TonesInDeepFreeze
The question I have though is, even if we don't have to worry about what it means to say the objects exist, haven't we just kicked the can down the road to the question "In what sense do the relations, the structures exist?"?
Buddhism was the original structuralism with their idea of utter dependence. Logic and the world, everything in fact, was dependent but not dependent on something
I'm not a set theorist, but I have some thoughts.
I haven't seen articles before that give a layman's explanation of forcing and of axioms for proving CH. So I appreciate that.
I have my own question. I kinda got the idea behind the explanation of the filter, but I wonder if this is correct:
That filter proves the existence of a certain real number. Then we use other filters to prove the existence of other real numbers. By doing that some uncountable number of ways, we prove that there are sets of real numbers that have cardinality between the cardinality of the set of natural numbers and the cardinality of the power set of the set of natural numbers. Is that correct?
(IMPORTANT. In this post, I take "ZF is consistent" as a background assumption. For example "ZFC is consistent" in this post means "If ZF is consistent then ZFC is consistent".)
'AC' stands for the axiom of choice.
'CH' stands for the continuum hypotheis.
'GCH' stands for the generalized continuum hypothesis.
'ZFC'stands for ZF+AC.
I know nothing about forcing other than this:
(1) Cohen used forcing to prove:
ZF+~AC.
ZFC+~CH is consistent. So, a fortiori, ZFC+~GCH is consistent.
(2) Forcing involves ultrafilters and/or Boolean algebra.
Re (1), Godel had previously proved:
ZFC is consistent. [*]
ZFC+GCH is consistent. So, a fortiori, ZFC+CH is consistent.
Godel did it with the notion of the constructible universe.
Combining Godel and Cohen, we get:
AC is independent from ZF.
CH is independent from ZF.
GCH is independent from ZF.
[*] But what about Godel's second incompleteness theorem that entails "Set theory does not prove its own consistency"? Well, actually the second incompletess doesn't entail that. The second incompleteness theorem does entail, "If set theory is consistent then set theory does not prove set theory is consistent". And, since I put "ZF is consistent" as a blanket assumption for this post, "ZFC is consistent" in this post, stands for "If ZF is consistent then ZFC is consistent". That qualification applies, mutatis mutandis, to both the Cohen theorems and both the Godel theorems above.
FOR REFERENCE:
df. A set of sentences S is consistent iff S does not prove a contradiction.
df. A sentence P is independent from a set of sentences S iff (S does not prove P and S does not prove ~P).
th. A sentence P is independent from a set of sentences S iff (S+P is consistent and S+~P is consistent.
th. If a set of sentences S is consistent, then there is a model in which every sentence in S is true (this is Godel's completeness theorem).
So:
To prove "ZFC is consistent", it suffices to prove there is a model of ZFC. But "ZF is consistent" entails "ZF has a model". So it suffices to prove that "ZF has a model" implies "ZFC has a model".
To prove "ZFC+GCH is consistent", it suffices to prove there is a model of ZFC+GCH. But "ZF is consistent" entails "ZF has a model", which, Godel proved, entails "ZFC has a model". So it suffices to prove that "ZFC has a model" implies "ZFC+GCH has a model".
To prove "ZF+~AC is consistent", it suffices to prove there is a model of ZF+~AC. But "ZF is consistent" entails "ZF has a model". So it suffices to prove that "ZF has a model" implies "ZF+~AC has a model".
To prove "ZF+~CH is consistent", it suffices to prove there is a model of ZF+~CH. But "ZF is consistent" entails "ZF has a model". So it suffices to prove that "ZF has a model" implies "ZF+~CH has a model".
Those are examples of relative consistency. "If theory T is consistent, then theory Y is consistent".
Some years after Godel's results just mentioned, Sierpenski proved that ZF+GCH proves ZFC. So:
Proving "ZF+GCH is consistent", a fortiori, proves "ZFC is consistent".
Proving "ZF+~AC is consistent", a fortiori, proves "ZF+~AC+~GCH", which, a fortiori, proves "ZF+~GCH is consistent". So, "ZF+~AC is consistent" proves "ZF+~GCH is consistent".
BACKGROUND:
What is the axiom of choice?
What is the continuum hypotheis?
What is the generalized continuum hypothesis?
df. 0 = the empty set.
df. PS = the set of subsets of S.
df. Y\Z = the set whose members are all and only those members of Y that are not members of Z.
C (the axiom of choice) is the statement:
"For every S, there is a function on the PS\{0} such that for every x in PS\{0}, f(x) e x". We call such an f "a choice function for S".
To visualize the above:
Imagine a nation made up of provinces (and possibly there are infinitely many provinces). From each province we can choose a representative who is a resident of that province.
If S is finite, without the axiom of choice, by a trivial induction on the cardinality of S, we prove there is a choice function for S, so we don't need the axiom of choice to prove there is a choice function for S.
"For every S, there is a function on the PS\{0} such that for every x in PS\{0}, f(x) e x".
The axiom of choice is equivalent with a number of other theorems, especially "Every S has a well ordering" and "Every S is equinumerous with an ordinal".
df. S^x = the set of funtions from x into S.
df. x and y are equinumerous iff there is a bijection between x and y.
df. card(S) = the least ordinal k such that S and k are equinumerous.
df. for ordinals, x < y iff x e y.
df. w = the set of natural numbers
df. R = the set of real numbers
th. card(R) = card(Pw) = card(2^w)
th. There is no surjection from S onto PS. (Cantor's theorem)
th. card(w) < card(R)
CH is the statement:
"There is no S such that card(w) < card(S) < card(R)".
GCH is the statement:
"If X is infinite, then there is no S such that card(X) < card(S) < card(PX).
Cantor failed to prove CH. Hilbert wanted somebody to prove it. Godel proved that we can't disprove GCH, a foritori that we can't disprove CH. Cohen proved that we can't disprove ZFC+~CH, so, a fortiori, we can't dispove ZFC+~GCH.
So some set theorists, who feel that ~CH fits their concept of 'set' have been trying to discover a set theoretic statement that is even more convincingly true to their concept of 'set' and that proves ~CH. Other set theorists, who feel that CH fits their concept of 'set' have been trying to discover a set theoretic statement that is even more convincingly true to their concept of 'set' and that proves CH.
FOR REFERENCE:
Z is basic infinitistic set theory (first order logic with identity, extensionality, schema of separation, pairing, union, power, infinity, regularity). ZF is Z with the axiom schema of replacement added.
The axiom schema of replacement is the statement [I'm simplifying]:
"If R is a function class, then, for every S, there is the T whose members are all and only those y such that there is an x in S such that <x y> in R."
R is a proper class. It is a proper class of ordered pairs. The axiom of replacement is: If R is functional (i.e. if <x y> in R and <x z> in R, then y=z), then for any set S, there exists the set T that is the image of S by R. I.e., if you have a set S, and a functional relation, then the "range" of that relation restricted from the domain S is a set.
That mentions proper a proper class, though Z proves ~Ex x is a proper class. So the actual axiom schema of replacement is a set of axioms, with each axiom mentioning a formula instead of a proper class. The formula "carves out" the proper class.
GCH.
If S is infinite, then is no set X such that all these hold:
S injects into X
There is no bijection between X and S
X injects into 2^S
There is no bijection between S and 2^S
And CH is just a special case of GCH, where S = w.
Two iconic books that handle forcing in detail are:
'Set Theory' - Jech
'Set Theory: An introduction To Independence Proofs' - Kunen
Jech's book is a great tome. But I might prefer Kunen, because, though I'm not a formalist, I find that a formalist sensibility contributes to good exposition.
However, both those books are at a graduate level. For an introduction to set theory, I always recommend:
Elements Of Set Theory - Enderton, which is a great read
supplemented with
Axiomatic Set Throry - Suppes.
I didn't know that, but it makes the problems which I've apprehended much more understandable. If everything is a set, in set theory, then infinite regress is unavoidable. A logical circle is sometimes employed, like the one mentioned here to disguise the infinite regress, but such a circle is really a vicious circle.
Quoting fishfry
I reject "the empty set" for a reason similar to the reason why I rejected a set with no inherent order. it's a fiction which has no purpose other than to hide the shortcomings of the theory. There are very good reasons why "0" ought to represent something in a class distinct from numbers. There are even reasons why "1" ought to be in a distinct class.
Quoting fishfry
This may be the case, but you ought to recognize that being elements of the same set makes them "the same" in a meaningful way. Otherwise, a set would be a meaningless thing. So when you said for instance, that {0,1,2,3,} is a set, there must be a reason why you composed your set of those four elements. That reason constitutes some criteria or criterion which is fulfilled by each member constituting a similarity.
Quoting fishfry
This is a simple feature of common language use. A word may receive its meaning through usage rather than through an explicit definition. That the word has no definition does not mean that it has no meaning, its meaning is demonstrated by its use, as is the case with an ostensive definition. Allowing that a word, within a logical system, has no explicit definition, allows the users of the system an unbounded freedom to manipulate that symbol, (exemplified by TonesInDeepFreeze's claim with "least"), but the downfall is that ambiguity is inevitable. This is an example of the uncertainty which content brings into the formal system, that I mentioned in the other thread.
Quoting fishfry
This I would say is a good representation of the philosophical concept of "infinite". Note that the philosophical conception is quite different from the mathematical conception. If every ordinal is a set composed of other ordinals, and there is no limit to the "amount" of ordinals which one may construct, then it ought to be very obvious that we cannot have an ordinal which contains all the ordinals, because we are always allowed to construct a greater ordinal which would contain that one as lesser. So we might just keep getting a greater and greater ordinal, infinitely, and it's impossible to have a greatest ordinal.
I think there is a way around this though, similar to the way that set theory allows for the set of all natural numbers, which is infinite. As you say, "set" has no official definition. And, you might notice that "set" is logically prior to "cardinal number". So all that is required is a different type of set, one which is other than an ordinal number, which could contain all the ordinals. It would require different axioms.
Quoting fishfry
This is not really true now, if we accept set theory. If "set" is logically prior to "number", then "set"
is a defining principle of "number". That is why you and I agreed that each ordinal is itself a set. We have a defining principle, an ordinal is a type of set, and a cardinal is a type of ordinal.
Quoting fishfry
Correction, at my worst I am a part-time Platonist. At my best I am a fulltime Neo-Platonist.
Quoting fishfry
We do not have to go the full fledged Platonic realism route here, to maintain a realism. This is what I tried to explain at one point in another thread. We only need to assume the symbol "5", and what the symbol represents, or means. There is no need to assume that the symbol represents "the number 5", as some type of medium between the symbol, and what the symbol means in each particular instance of use. So when I say that a thing exists, and has a measurement, regardless of whether it has been measured, what I mean is that it has the capacity to be measured, and there is also the possibility that the measurement might be true.
Quoting fishfry
If you think that I was advocating for mathematical Platonism, then you misunderstood. I was advocating for realism.
Quoting fishfry
A mathematical Platonist thinks of ideas as objects. I recognize the reality of ideas, and furthermore I accept the priority of ideas, so I am idealist. But I do not think of ideas as objects, as mathematical Platonists do, I think of them as forms, so I'm more appropriately called Neo-Platonist.
Quoting fishfry
This is that vague boundary, the grey area between fact and fiction which we might call "logical possibility". If we adhere to empirical principles, we see that there are individual objects in the world, with spatial separation between them. If we are realist, we say that these objects which are observed as distinct, really are distinct objects, and therefore can be counted as distinct objects. We might see three objects, and name that "3", but "3" is simply what we call that quantity. Being realist we think that there is the same quantity of objects regardless of whether they've been counted and called "3" or not.
But if we give up on the realism, and the empirical principles, there is no need to conclude that what is being seen is actually a quantity of 3. There might be no real boundaries between things, and anything observed might be divisible an infinite number of times. Therefore whatever is observed could be any number of things. This is the world of fiction, which some might call "logical possibility", and you call pure mathematics. Empirical truths, like the fact that distinct objects can be counted as distinct objects, pi as the ratio between circumference and diameter of a circle, and the Pythagorean theorem, we say are discovered. Logical possibilities are dreamt up by the mind, and are in that sense fictions.
I do not mean to argue that dreaming up logical possibilities is a worthless activity. What I think is that this is a primary stage in producing knowledge. We look at the empirical world for example and create a list of possibilities concerning the reality of it. The secondary stage is to eliminate those logical possibilities which are determined to be physically impossible through experimentation and empirical observation. So we proceed by subjecting logical possibilities, and axioms of pure mathematics, to a process of elimination.
Of course the reals consist of rationals and irrationals. That's provable from the axioms. Every model of the reals satisfies the axioms of the reals. FWIW I'm familiar with the work of Natalie Wolchover, the author of the article you linked. She puts the "pop" in pop science; which is to that that she's very good, up to a point; but not past that point. I didn't read the article and can't vouch for anything she might have said.
I can't really describe forcing. Timothy Chow, the author of A Beginner's Guide to Forcing, describes forcing as an "open exposition problem." That is, just as an open problem is a problem nobody knows how to solve, forcing is a subject that nobody knows how to explain to non-specialists.
I strongly recommend his article for anyone interested in the subject; bearing in mind that nobody would be expected to understand much of it, and the more times you read it and the more you read about forcing in general, the better vague understanding you'll get. But there's no known explanation that's any easier than diving in and learning the actual set theory, and it's a notoriously difficult subject.
The basic idea is that we want to know what things are consistent with a given set of axioms, so we try to find models that satisfy the axioms and also satisfy the extra things we're interested in. For example in geometry we can take the Euclidean axioms minus the parallel postulate (PP). We know the PP is consistent because Euclidean geometry is a model of the axioms plus PP. On the other hand in the 1840's, Riemann and others discovered that the axioms plus not-PP also had a model. This means that you can take the other axioms with PP or with the negation of PP, and both resulting systems are consistent. Alternatively, you can say that PP is "independent" of the other axioms; given the axioms, you can neither prove nor disprove PP.
A slightly more sophisticated example if you've seen group theory is that if you take the axioms for a group, you might want to know whether the "Abelian axiom" is consistent and/or provable; namely, is it true that for all x and y in a group, xy = yx.
Well, the integers with addition are an example of a group in which it's true. But the set of invertible 2x2 matrices with multiplication also form a group, and there are examples where commutativity fails. Since there are models of the group axioms with ah]d without commutativity, we would say that the "Abelian axiom" is independent of the group axioms.
Ok. Now with that in mind, down to cases. Cantor called the cardinality of the natural numbers [math]\aleph_0[/math]. He proved that the cardinality of the real numbers was [math]2^{\aleph_0}[/math]. And he showed that the next larger cardinal after [math]\aleph_0[/math] is [math]\aleph_1[/math].
So we have [math]\aleph_0[/math] directly followed by [math]\aleph_1[/math]. And out there among the Alephs is [math]2^{\aleph_0}[/math]. The question is, might it be the case that [math]2^{\aleph_0} = \aleph_1[/math]? This question, or rather the claim that equality holds, is the continuum hypothesis (CH).
Cantor was unable to prove CH, and neither was anyone else. In 1940 Kurt Gödel proved that at the very least, CH was consistent with the other axioms of ZF, Zermelo-Fraenkel set theory. He did this by exhibiting a model in which it was true. This model is called Gödel's constructible universe. It's a universe of sets in which all the axioms of ZF are true, and in which CH is true. This showed that at the very least, assuming CH did not introduce any contradiction into ZF that wasn't already there.
That last remark needs explanation. What do I mean that CH doesn't introduce a contradiction that wasn't already there? Recall that Gödel had already proven in 1931 that ZF can't prove itself consistent. So the only way to know if ZF is consistent is to introduce even stronger principles that in effect assume it is. For all we know, set theory is inconsistent.
What Gödel showed, then, is that if ZF is consistent, so is ZF + CH. That is, all these proofs are relative consistency proofs. They don't show that anything is consistent; they only show that IF one system is consistent, then so is that system plus some other stuff.
But what about the negation of CH? Is that consistent with ZF as well? Gödel had shown that there's a model of ZF + CH. Could there be a model of ZF + not-CH? The problem is that nobody had any idea how to cook up alternative models of ZF. This was a real problem.
In 1963 an analyst named Paul Cohen figured it out. By analyst,I mean he was into real analysis -- epsilons and deltas and convergence and such. About as far away from mathematical logic as you can get. He woke up one day and said to himself, "I think I'll take a run at CH." He figured out how to cook up alternative models of ZF. In 1966 he won the Fields medal, the only Fields medal ever given for mathematical logic. I have always assumed that all the other official professional logicians must have been mighty annoyed. Some nonspecialist wakes up one day and solves the greatest unsolved problem in your field.
So ok all of that is preamble. How did he cook up alternative models? He invented a method called forcing. And having come this far, I really can't say much about it; first, because I don't know much about it myself, and second, because as Timothy Chow noted, nobody knows how to explain this to nonspecialists.
The idea basically is analogous to the procedure in abstract algebra where we adjoin roots to fields. That is, suppose that we believe in the rational numbers. We know the rationals satisfy the field axioms: you can add and multiply rationals to get another rational. Multiplication distributes over addition. And every nonzero rational has a multiplicative inverse.
Now suppose we want to prove that there is a field that contains the rationals and that also contains the square root of 2. We "adjoin" a meaningless symbol, [math]\sqrt 2[/math], to the rationals. We know nothing about this symbol other than that it has the formal property [math](\sqrt{2})^2 = 2[/math].
In order to preserve the field axioms we have to say that all possible additions and multiplications are also in our new "extension field," as it's called. So we have a set of expressions of the form [math]a + b \sqrt 2[/math]. We can then prove that the resulting system of formal expressions [math]a + b \sqrt 2[/math] itself satisfies the field axioms.
This is the best analogy for forcing. We start with a model of set theory, and we carefully add new "thingies," whatever they are, making sure that our new system also satisfies the axioms of set theory. If we're clever, we can arrange things so that CH turns out to be false in our new model. Then we collect our Fields medal. Cohen was clever.
So the idea -- and this is pretty much everything I know about it -- is first, we start with a model of ZF. But wait, since we can't prove within set theory that set theory is consistent, we don't know for sure if there even IS a model of set theory. But no worries. If there is no model, then set theory is inconsistent, and then we can prove ANYTHING, including CH and its negation. So, to get things off the ground, we assume that set theory is consistent and that it has a model.
Then -- and this step comes out of nowhere, pretty much -- since there is a model, there is a countable model. This is the famous, or infamous, Löwenheim–Skolem theorem.
What on earth does it mean to have a countable model of set theory? Doesn't ZF prove that there are uncountable sets? Well yes, it does. But now we have to broaden our understanding of what that means. What does it mean for a set to be countable? It means there is a bijection from that set to the natural numbers.
Suppose we have some set X, and a bijection from X to the naturals. So X is countable. Now suppose we have some model of set theory, and we throw out all the bijections between X and the naturals from the model, making sure we still have a model. Then from "inside the model," X would be uncountable; but from outside the model, from our God-like perspective, we can perfectly well see that X is really countable.
So we learn that uncountability is a "relative" notion. A set may be countable in one model, and uncountable in another. It just depends on which bijections are lying around.
So we assume we have a model of ZF; and we then know that if we do, there must be a countable model; and then we can show that if there's a countable model, there is a countable, transitive model.
Having done that, we take a countable, transitive model of ZF, and carefully add sets to it, making sure that we preserve the axioms of ZF, while cooking up a violation of CH.
Well there you go. I wrote a lot of words and didn't explain a thing about what forcing is. Definitely go read Tim Chow's excellent article.
Since no two things are identical in spatiotemporal reality, do you also reject the number 2?
Thank you. That's very clear. I appreciate your commentary.
Quoting fishfry
She popped Cohen's Filter pretty good. :smile:
Quoting TonesInDeepFreeze
And good ones they are! Thank you for your commentary.
Sets can contain other sets. In fact a set is "something" in addition to its constituent elements. It's a "something" that allows us to treat the elements as a single whole. If I have the numbers 1, 2, and 3, that's three things. The set {1,2,3} is one thing. It's a very subtle and profound difference. A set is a thing in and of itself. The empty set, the set containing the empty set, and the set containing the set containing the empty set are three distinct sets. Which we could, if we wanted to, take together into a set! Like this:
[math]\{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}\}[/math].
Quoting Gregory
Zero is not nothing. Zero is a particular point on the real number line. Or the address or location of a point, if you prefer. Zero is a particular thing. It represents the cardinality of the set of purple flying elephants in my left pocket. Zero is something. Nothing is nothing. Which can be read two different ways!
Quoting Gregory
Well, what's a process? If a process is something that can be "contained," it needs explanation.
Quoting Gregory
The natural numbers seem permanent. They don't change from day to day, whether you regard them as sets, as in the von Neumann finite ordinals, or as a process of starting at 0 and taking successors.
I am no expert on these things though.
Quoting jgill
You're very welcome, glad that helped.
You write very well. That must be why I like to engage with you, not that I want to troll you.
Quoting Luke
Yes that' the mathematical Platonism I reject. I believe we had a lengthy discussion on this in the other thread, you and I. The number 2 is an unnecessary intermediary between the symbol, and what the symbol represents, or means, in use. Of course you might use the symbol "2" to represent the number 2, but then you are writing fiction.
Quoting fishfry
This is what I was asking about earlier, what allows for that unity if not some judgement of criteria, making the elements similar, or the same in some respect., a definition. This is a very important ontological question because we do not even understand what produces the unity observed in an empirical object.
Suppose you arbitrarily name a number of items and designate it as a set. You have created "a thing" here, a set, which is some form of unity. But that unity is completely fictitious. You are just saying that these items compose a unity called "a set", without any justification for that supposed unity. In its simplest from, this is the issue of counting apples and oranges. We can count an apple and orange as two distinct objects, and call them 2 objects. But if we want to make them a set we assume that something unifies them. If we are allowed to arbitrarily designate unity in this way, without any criteria of similarity, then our concept of unity, which some philosophers (Neo-Platonist for example) consider as fundamental loses all its logical strength or significance.
How can either the number 2 or the numeral "2" represent or mean anything in use if no two things are identical in spatiotemporal reality? Isn't the law of identity the basis of your mathematics?
Thank you. I'll get to your second post later, I'm falling behind.
Quoting Metaphysician Undercover
Yes. Everything is a set. Or what they call a "pure set," meaning a set whose elements are also sets. There are as I mentioned set theories with urlements, also called atoms, but these are niche theories and not of interest to us at present. So everything is a set that contains other sets.
Quoting Metaphysician Undercover
No not at all. First, what's wrong with infinite regress? After all the integers go backwards endlessly: ..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ... You can go back as far as you like. I'm fond of using this example in these endlessly tedious online convos about eternal regress in philosophy. Cosmological arguments and so forth. Why can't time be modeled like that? It goes back forever, it goes forward forever, and we're sitting here at the point 2021 in the Gregorian coordinate system.
However in set theory there is no infinite regress. That's guaranteed by the axiom of foundation, also known as the axiom of regularity. It says that no set is a member of itself and it also rules out all circular membership chains like [math]a \in b \in c \in a[/math] and so forth. In standard set theory all sets are well-founded. That means that if you take its elements, which are themselves sets; and take their elements, which are themselves sets; and drill all the way down; you are guaranteed to hit bottom. There is no possible infinite regress of sets.
For completeness I'll mention that people do study non well-founded sets, but this is yet another niche interest and of no interest to us here. In standard set theory all sets are well-founded. There can never be an infinite regress of sets.
Quoting Metaphysician Undercover
Oh jeez man, you embarrassed yourself a little here. See now I feel bad pointing out that you embarrassed yourself because you complimented me. LOL.
@jgill was referring to the Riemann sphere, a way of viewing the complex numbers as a sphere. It's based on the simple idea of stereographic projection, a map making technique that allows you to project the points of a sphere onto a plane. There is nothing mystical or logically questionable about this. You should read the links I gave and then frankly you should retract your remark that the Riemann sphere is a "vicious circle." You're just making things up. Damn I feel awful saying that, now that you've said something nice about me.
Quoting Metaphysician Undercover
I find this deeply inconsistent with other things you've said. Earlier I was making the point that we can have two sets, X and Y, with a bijection between them, and we can say they are "cardinally equivalent," without knowing what that exact cardinal number is. Then later we can define cardinal numbers, and assign one of them to X and Y.
You claimed that the cardinal numbers were already "out there" waiting to be assigned. You used that idea to claim that I was wrong about ordinals being logically prior to cardinals.
So you somehow manage to believe in the existence of cardinal numbers, which include the endless hierarchy of gigantic cardinals given to us by Cantor's theorem: that a set's powerset is always of a strictly larger cardinality than the set. So we have the cardinality of the natural numbers, which is smaller than the powerset of the natural numbers, which is smaller than the powerset of the powerset of the natural numbers, and on and on forever.
You believe in the metaphysical existence of all of these humongously unimaginable cardinals; yet you deny the existence of the empty set on which they're all founded.
That's logically inconsistent.
But never mind that. I don't believe in the existence of the empty set either. Not in reality. If I see a table with nothing on it, there's nothing on it. I do NOT see the empty set sitting on the table. So I agree with you, I don't believe in the empty set.
But I DO believe in the empty set as a formal construction in the game of math. In fact the empty set is the extension of the predicate [math]x \neq x[/math]. Surely you must agree with that, since you believe in the law of identity.
Can you clarify your remark? If you don't believe in the metaphysical existence of the empty set, I'm in complete agreement. But if you claim to disbelieve in the empty set as a mathematical object, that's like disbelieving in the way the knight moves in chess. You can't disbelieve in it, it's just one of the rules of the game.
And again; if you are so strong on the law of identity, then you must believe that [math]\emptyset = \{x : x \neq x \}[/math].
Quoting Metaphysician Undercover
I don't know what you mean? What do you mean by "class" in this context? Is 2 in its own class? Why not, it's 1 + 1, right? Although in the past you've denied even that, so I hope we're not going down that road again.
Quoting Metaphysician Undercover
Jeez Louise man. I say: "The only thing they have in common is that they're elements of a given set." And then you say I "ought to recognize ..." that very thing.
Did you simply not read what I wrote? Do you like to just push my buttons? I say something as clear as day; and you respond by admonishing me that I "ought to recognize" the very thing I've just said. I don't get it. That's why I sometimes think you are trolling me.
Quoting Metaphysician Undercover
Ok, {5, my lunch, the Mormon Tabernacle Choir}. What of it?
Quoting Metaphysician Undercover
A very disingenuous point. The elements of a set need have no relation to one another nor belong to any articulable category or class of thought, OTHER THAN being gathered into a set.
Quoting Metaphysician Undercover
Ok, you are now agreeing with me on an issue over which you've strenuously disagreed in the past. You have insisted that "set" has an inherent meaning, that a set must have an inherent order, etc. I have told you many times that in set theory, "set" has no definition. Its meaning is inferred from the way it behaves under the axioms.
And now you are making the same point, as if just a few days ago you weren't strenuously disagreeing with this point of view.
But in any event, welcome to my side of the issue. Set has no definition. Its meaning comes exclusively from its behavior as specified by the axioms.
Quoting Metaphysician Undercover
Completely agreed. And therefore a mathematical set has no inherent order, because [i]that's how sets are used in set theory[i]. Can you see that you've now completely conceded the point?
Quoting Metaphysician Undercover
No question about it. There are philosophers and set theorists who question whether the mathematical conception of set is even coherent. You'll get no disagreement from me on that point. Although just because @Tones forgot to mention that ordinal < means set membership doesn't support your point, it only means @Tones forgot to mention it.
Quoting Metaphysician Undercover
No question. As Hilbert famously pointed out about the axioms of geometry, "One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs."
That's a perfect expression of the formalist position on axiomatic systems.
Quoting Metaphysician Undercover
Yes, very good! That's essentially the proof. Any set of ordinals is itself an ordinal. Hence there is no set of all ordinals.
Quoting Metaphysician Undercover
Cesare Burali-Forti couldn't have said it better himself.
Quoting Metaphysician Undercover
The collection of all ordinals is a proper class. In standard set theory, ZF or ZFC, there are no official proper classes, so "proper class" is a colloquial expression. There are set theories in which proper classes are formalized. Either way, a proper class is a collection that's too big to be a set. The class of all sets, the class of all ordinals, etc.
Quoting Metaphysician Undercover
I am disappointed that you didn't accept the historical point I made earlier. Zero, negative numbers, irrational numbers, complex numbers, and transfinite numbers didn't used to be accepted as numbers, and now they are. Likewise p-adic and quaternions, two other types of numbers discovered only in the past couple of centuries. "Number" is a historically contingent concept.
Quoting Metaphysician Undercover
Not at all. Bricks are the constituents of buildings, but all the different architectural styles aren't inherent in bricks. There are plenty of sets that aren't numbers. Topological spaces aren't numbers. The set of prime numbers isn't a number. Groups aren't numbers. The powerset of the reals isn't a number. Just because numbers are made of sets in the formalism doesn't mean every set is a number.
Quoting Metaphysician Undercover
So now you agree that ordinals are logically prior to cardinals? I am glad you have internalized this fact to the point where it now seems obvious to you, when only a few days ago you were strenuously disagreeing.
But so what? It's true that in set theory everything is a set, but that doesn't mean everything is a number. I don't follow your logic.
Quoting Metaphysician Undercover
I looked that up on Wikipedia and it seemed to be about some kind of mystical emanation from "The One." Lost me, I'm afraid. But I'm shocked that you believe in the vast multitude of gigantic cardinal numbers, while professing disbelief in the empty set.
Quoting Metaphysician Undercover
I don't doubt that you tried to explain this to me and I missed it. Even now I don't think I know the difference between Platonism and realism.
Quoting Metaphysician Undercover
I'm afraid you lost me a bit there. The number 5 exists as a formal symbol and concept in set theory. What it is "for real" I am not sure. It's the thing that comes after 4, that's the structuralist idea, I think.
Quoting Metaphysician Undercover
Ok. I admit to being unclear on this. I'm only struck by finding you believing in the pre-existence of the vast array of cardinal numbers, yet disbelieving in the empty sets and set theory in general.
Quoting Metaphysician Undercover
Ok. Is this a bit structuralist? Natural numbers aren't particular things, but they are the relations among them; that is, 5 is the thing that follows 4, and that's all I need to know about it.
Quoting Metaphysician Undercover
Ok. But 3's easy. How about the transfinite cardinals? You believe in them yet disbelieve in set theory? That's a hard row to hoe.
Quoting Metaphysician Undercover
You are now willing to agree with me that there may be some virtue to considering math to be an interesting and useful fiction? @Meta I find you agreeing with my point of view in this post.
Quoting Metaphysician Undercover
You are mellowing! And agreeing with me!! I must be having an effect. I will say that you have achieved some genuine mathematical insight lately.
Quoting Metaphysician Undercover
So you would ban the teaching of Euclidean geometry now that the physicists have accepted general relativity? We disagree on this. Math is the study of that which is logically possible. Math leaves what's real to the physicists. And of course even the physicists no longer have much interest in what's real, but that's a criticism for another time. But math is not bound by what's real. On this we disagree strongly.
Quoting Metaphysician Undercover
Would you ban Euclidean geometry from the high school curriculum because it turns out not to be strictly true?
There is no criterion. In fact there are provably more sets than criteria. If by "criterion" you mean a finite-length string of symbols, there are only countably many of those, and uncountably many subsets of natural numbers. So most sets of natural numbers have no unifying criterion whatsoever, They're entirely random.
Quoting Metaphysician Undercover
I just proved that most sets of natural numbers are entirely random. There is no articulable criterion linking their members other than membership in the given set. There is no formal logical definition of the elements. There is no Turing machine or computer program that cranks out the elements. That's a fact.
Quoting Metaphysician Undercover
Ok.
Quoting Metaphysician Undercover
Tru dat. Or as the kids say, Yes, indubitably so.
After all as I just noted, there are only countably many criteria, formulas, computer programs. But there are uncountably many sets of natural numbers. Most of those sets are entirely random. There is no rhyme or reason to their constituent members. They're just random collections.
Unless you are a constructivist, in which case you deny the existence of random sets. Some go down that path.
Quoting Metaphysician Undercover
Only their collection into a set.
Quoting Metaphysician Undercover
Do you deny the existence of all sets that cannot be cranked out by a Turing machine or at least defined in first-order logic? You can do that if you like. I don't see the use. Consider the following thought experiment. You flip a fair coin a countably infinite number of times. You thereby generate a sequence of 1's and 0's. What invisible magic forces the resulting bitstring to be computable, or describable by an algorithm or formula? Why can't the result be completely random, having no pattern at all? That's by far the most likely outcome.
We define the set of symbols prior to a symbol s:
If s is primitive, prior(s) = 0
If s is defined, prior(s) = U{k | Et(t occurs in the definiens for s & k = prior(t))}
Then:
j is prior to s <-> j e prior(s)
Obviously, "2" refers to two distinct and different things. If there was only one thing we'd have to use "1".
Quoting fishfry
Again, this is the difference between fiction and fact. We can imagine infinite regress, and imagine time extending forever backward, but it isn't consistent with the empirical evidence. That's the problem with infinite regress, it's logically possible, but proven through inductive (empirical) principles (Aristotle's cosmological argument for example) to be impossible.
Quoting fishfry
I beg to differ. Didn't we go through this already in the Gabriel's horn thread. It seems like you haven't learned much about the way that I view these issues. You write very well, but your thinking hasn't obtained to that level. Another example of the difference between form and content.
Quoting fishfry
Are you denying the contradiction in what you wrote? If they are members of the same set, then there is a meaningful similarity between them. Being members of the same set constitutes a meaningful similarity. You said "the elements of a set need not be 'the same' in any meaningful way. The only thing they have in common is that they're elements of a given set." Can't you see the contradiction? If they are said to be members of the same set, then they are the same in some meaningful way. It is contradictory to say that they are members of the same set, and also say that they are not the same in any meaningful way.
Another example of this same sort of contradiction is when people refer to a difference which doesn't make a difference. If you apprehend it as a difference, and speak about it as a difference, then clearly it has made a difference to you. Likewise, if you see two things as elements of the same set, then clearly you have apprehended that they are the same in some meaningful way. To apprehend them as members of the same set, yet deny that they are the same in a meaningful way, is nothing but self-deception. Your supposed set is not a set at all. You are just saying that there is such a set, when there really is no such set. You are just naming elements and saying "those are elements of the same set" when there is no such set, just some named elements. Without defining, or at least naming the set, which they are members of, there is no such set. And, naming the set which they are elements of is a designation of meaningful sameness.
Here is a feature of imaginary things which you ought to learn to recognize. I discussed it briefly with Luke in the other thread. An imaginary thing (and I think you'll agree with me that sets are imaginary things, or "pure abstraction" in your terms) requires a representation, or symbol , to be acknowledged. And, for an imaginary thing, to exist requires being acknowledged. However, the symbol, or representation, is not the imaginary thing. The imaginary thing is something other than the symbols which represent it. So the imaginary thing necessarily has two distinct aspects, the representation, and the thing itself, the former is called form, the latter, content. And this is necessary of all imaginary things.
The important point is that you cannot claim to remove one of these, from the imaginary thing, because both are necessary. So a purely formal system, or pure content of thought, are both impossibilities. And when you say "these things are elements of the same set", you have in a sense named that set, as the set which these things are elements of, thereby creating a meaningful similarity between them. The point being that a meaningful similarity is something which might be created, solely by the mind and that is how the imagination works in the process of creating fictions. But when something is a creation, it must be treated as a creation.
Quoting fishfry
Again, incoherency fishfry. Can't you see that? There is necessarily a reason why you place them in the same set, and this 'reason why' is something other than actually being in the same set. You are not acknowledging that "being gathered into a set" requires a cause, and that cause is something other than being in the same set. So the relation that the things have to one another by being in the same set is not the same as the relation they have to one another by being caused to be in the same set. And things which are in the same set necessarily have relations to each other which are other than being in the same set, because they have relations through the cause, which caused them to be in the same set.
A set is an articulable category, or class of thought! If a set is not a class of thought, then what the heck is it, jeez louise? And don't tell me it might be anything because it is not defined, because even "anything" is a class of thought.
Quoting fishfry
It appears like you didn't read what I said. That a word is not defined does not mean that it has no meaning. As I said, it may derive meaning from its use. If the word is used, then it has meaning. So if "set" derives it's meaning from the axioms, then there is meaning which inheres within, according to its use in the axioms.
Quoting fishfry
What we do not agree on is what "inherent order" means. i really do not see how you get from the premise, that "set" is not defined, but gets its meaning from its use, to the conclusion that a set might have no inherent order. In order for the word "set" to exist, it must have been used. Therefore it is impossible for "set" not to have meaning, and we might say that there is meaning (order, if order is analogous to meaning, as you seem to think), which inheres within. Wouldn't you agree with this, concerning the use of any word? If the word has been used, there is meaning which inheres within, as given by that use. And, for a word to have any existence it must have been used.
Quoting fishfry
It appears like you misunderstood. I didn't say every set is a number, to the contrary. I said that if we proceed under the precepts of set theory, every number is a set. Therefore we cannot say that "number" is undefined because "set" is now a defining feature of "number", just like when we say every human beings is an animal, "animal" becomes a defining feature of "human being".
Quoting fishfry
Didn't it strike you that I was in a very agreeable mood that day? Now I'm back to my old self, pointing out your contradiction in saying that things could be in the same set without having any meaningful relation to each other, other than being in the same set. You just do not seem to understand that things don't just magically get into the same set. There is a reason why they are in the same set.
Maybe at some point we'll discuss the supposed empty set. How do you suppose that nothing could get into a set?
Quoting fishfry
Actually I do not agree with general relativity, so I would ban that first.
Quoting fishfry
You keep saying things like this, the Pythagorean theorem is not true, now Euclidian geometry in general is not true. I suppose pi is not true for you either? Until you provide some evidence or at least an argument, these are just baseless assertions.
Quoting fishfry
On what basis do you say they are a unity then? You have a random group of natural numbers. Saying that they are a unity does not make them a unity. So saying that they are a "set" does not make them a unity. This is where you need a definition of "set" which would make a set a unity.
Quoting fishfry
Then you have no basis to your claim that a set is a unity. And you cannot treat a set as a unified whole. If a set is supposed to be a unified whole, then you cannot claim that "set" is not defined.
I'm missing your point also. What's your gripe about the innocuous Riemann sphere? :chin:
"2" can also refer to two distinct but same things, such as "things" of the same type or category. But all categories/classifications are equally as fictitious and man-made as the sets and orders you reject. As you say:
Quoting Metaphysician Undercover
Scientists justified both the inclusion and exclusion of Pluto as a planet at different times. Like Pluto, many individual "things" are borderline cases in their classification. Moreover, nothing guarantees the perpetuity of any category/set, or of what defines ("justifies") the inclusion of its members.
Furthermore, if you base your mathematics on empiricism rather than on "abstraction" or "fiction", then you must also reject fractions, since a half cannot be exactly measured in reality.
Quoting Metaphysician Undercover
If there are "no real boundaries between things", then acknowledging that "anything observed might be divisible an infinite number of times" is not to "give up on the realism", but to adhere to it.
To make infinite numbers into a circle is to make a vicious circle. It is to say that the beginning is the same as the end. And this is what allows for the faulty view of time which fishfry described.
Quoting Luke
This is a different sense of "same", not consistent with the law of identity.
Quoting Luke
When they are based in empirical observation they are not equally fictitious. Remember, fishfry speaks of pure abstraction, and claims that a set might be absolutely random..
Quoting Luke
That a person later decides to have been wrong in an earlier judgement, is not relevant.
Quoting Luke
I do reject fractions, I believe that the principles employed are extremely faulty, allowing that a unit might be divided in any way that one wants. This faultiness I believe, is responsible for the Fourier uncertainty In reality, how a unit can be divided is dependent on the type of unit.
Quoting Luke
That's the case if there are "no real boundaries between things". But I am arguing that empirical evidence demonstrates that there are real boundaries.
So "2" cannot refer to two distinct but same things? You cannot have 2 apples or 2 iPhones, etc?
Quoting Metaphysician Undercover
So "2" can refer to two distinct but same things? You can have 2 apples or 2 iPhones, etc?
Quoting Metaphysician Undercover
The categories we use are either discovered or man-made. If they are discovered, then how can we be "wrong in an earlier judgement" about them; why are there borderline cases in classification; and why does nothing guarantee their perpetuity as categories?
Quoting Metaphysician Undercover
You need help.
Quoting Metaphysician Undercover
What principles should be employed?
Quoting Metaphysician Undercover
How many slices should a cake or a pizza have? Also, doesn't this reintroduce the fractions you rejected?
Quoting Metaphysician Undercover
Where's the argument?
How can one argue about this? It is so silly. :lol:
So the circles vicious as opposed to innocuous?
:cool:
That's the world that MU lives in. :roll:
But we're not talking "fact," if by that you mean the real world. The subject was set theory, which is an artificial formal theory. Set theory is not any part of any physical theory. I pointed out to you that in set theory, everything is a set, including the elements of sets. You responded by saying you hadn't realized that. I thought we were therefore making progress: You acknowledged learning something you hadn't known before. And now you want to revert back to "fact," as if set theory has an ontological burden. It does not.
Quoting Metaphysician Undercover
But I never claimed it did. I offered the mathematical example of the integers. Are you a disbeliever in sufficiently small negative numbers? Do you believe in -47? -48? -4545434543? Where does your belief stop? Of course this is not a physical example, it's a mathematical example; in fact, an example that illustrates the difference between physics and math.
Quoting Metaphysician Undercover
Ok! Then we are in agreement. Since I have made absolutely no other claims. So just to satisfy my curiosity, do you believe in the negative integers? They believe in you.
Quoting Metaphysician Undercover
Discussion for another time, but I have made no claims about the world. Why do you argue as if I did?
Quoting Metaphysician Undercover
That was a lengthy thread from a while ago. Can you remind me of the specifics? It's not possible that "we went through this" about the Riemann sphere. Stereographic projection is a commonplace idea among every mapmaker since antiquity who's wrestled with the dilemma of representing a spherical earth on a flat map. Can you remind me of what on earth you might be talking about? You place a sphere above a plane. From the north pole of the sphere, you draw a straight line through a point on the sphere and extend the line to a point on the plane. You thereby have a mapping from the sphere to the plane. In cartography it's a basic technique. In complex variables theory, it's a way of visualizing the complex numbers as a sphere. There is no mysticism or "vicious circle" or any such nonsense as you claim.
Quoting Metaphysician Undercover
If you would say what you're talking about, I can respond. The Gabriel's horn thread was lengthy and long past. Tell me what you're talking about.
In any event, I've learned far too much about how you view things.
Quoting Metaphysician Undercover
Your ignorance is only matched by your ill manners. Going forward, if you can't be civil, put a sock in it.
Quoting Metaphysician Undercover
I repeatedly said that the only thing they have in common is being elements of the given set. So why are you acting like I haven't said that every single time?
Quoting Metaphysician Undercover
Only in a sophistic sense. I already pointed out to you that if "meaningful similarity" or "property" or "predicate" is interpreted as referring to an idea expressible in a finite-length string of symbols, there are more subsets of the natural numbers than there are properties. Therefore most sets are entirely random. Their elements have nothing at all in common except for being gathered into the given set.
Quoting Metaphysician Undercover
Ok fine, on that definition. I'll agree. But it's a pretty trivial point. Especially for you to be going on about it.
Quoting Metaphysician Undercover
Other than being in the same set. You deliberately quote me out of context to make a point. Disingenuous much?
Quoting Metaphysician Undercover
No. What I said is perfectly accurate.
Quoting Metaphysician Undercover
Only that they are members of the same set. So what? You are being childish to go on like this.
Quoting Metaphysician Undercover
Yeah yeah.
Quoting Metaphysician Undercover
Why are you going on like this? Let me remind you of the conversation. You expressed realization that in set theory, everything's a set. Then you claimed that leads to infinite regress. I pointed out that one, there's nothing logically wrong with infinite regress. I gave the negative integers as an example.
Then I pointed out that in set theory, we adopt the axiom of foundation to explicitly rule out infinite regress. You totally ignored both those points to go off on this trivial and pointless tangent.
Quoting Metaphysician Undercover
Well, for sake of discussion, it's not clear to me that every imaginary thing has a referent. Sets, for example. The empty set is an imaginary formal thing, but I don't know that it has a referent. Certainly not in the physical world.
Quoting Metaphysician Undercover
Focus. Focus. You said that the fact that in set theory everything is a set, leads to infinite regress. I pointed out that the negative integers are an example of an unproblematic negative regress; and that the axiom of foundation rules out infinite regresses of set membership.
You have avoided both those points to go off on trivialities and irrelevancies. And personal insults. What's the point?
Quoting Metaphysician Undercover
Oh no, not at all. The powerset of the natural numbers is uncountable. There are more sets than reasons. Most sets have no reason at all.
You've gone from saying that the elements have something in common, namely being in the given set -- which I agree with -- to now saying that there's some OTHER reason in addition to that. You're simply wrong about that. The powerset of the natural numbers exists, that's an axiom of set theory. Every set has a powerset, the set consisting of all the set's subsets. And the powerset is far larger than the set itself. There aren't enough "reasons" or predicates or explanations to cover them all, by a countability argument.
Quoting Metaphysician Undercover
You're thinking of the south and the Civil war. A side in a war needs a cause. A set needs no cause. Show me in the axioms for set theory where it says that. This is just something you made up. Again, you're trying to reify sets; but sets are only imaginary formal entities whose behavior is entirely determined by the axioms.
Quoting Metaphysician Undercover
You're just making that up. And changing the subject.
I challenged you on your claim that the idea that sets contain only other sets leads to infinite regress. I pointed out that the axiom of foundation precludes infinite regress of set membership. You changed the subject.
Quoting Metaphysician Undercover
You can say the knight flies over the moon, but that's not in the rules of chess. There are no "causes" in the axioms of set theory. So you're just making this up and then typing in crap, and wasting my time trying to get you to focus on the actual conversation we were having, which for a brief moment got substantive before you reverted to just making things up.
Quoting Metaphysician Undercover
There are not enough predicates to cover all the sets that there are. Most sets have no reason or cause at all; they're pure randomness.
Quoting Metaphysician Undercover
I could say the same about you. But I have read what you've said. What you've said is wrong; and your repeating it doesn't make it any less wrong.
Quoting Metaphysician Undercover
Ok. Fine. But there are no "causes" in the axioms.
Quoting Metaphysician Undercover
Don't start that crap again. I can't help it if you reject modern math. I can't do anything about that.
Quoting Metaphysician Undercover
You've worn me out. I'm losing interest.
Quoting Metaphysician Undercover
Well as Bill Clinton said, that depends on what the meaning of "is" is. If you mean that a number literally is a set, no, that's not true, as Benacerraf so insightfully pointed out. If you mean that in set theory a number is represented by a set, then that's true. Important for you to make that distinction.
Quoting Metaphysician Undercover
In set theory, a number is defined as a particular type of set. Just because set is an undefined term doesn't mean that we can't use it to define other things. Just as point is an undefined term in Euclidean geometry, but a line is made of points. Right? Right.
Quoting Metaphysician Undercover
Yes, that didn't last long. But you were more than agreeable the other day. You actually achieved some insight. You realized that a set has no definition, and that its meaning is derived from the axioms. You realized that the members of sets are also sets.
Quoting Metaphysician Undercover
You're just wrong about that. Provably wrong, since there aren't enough reasons to cover uncountably many sets.
Quoting Metaphysician Undercover
By the axiom of pairing, which has as a consequence the fact that if [math]X[/math] is a set, so is [math]\{X\}[/math]. Everything's given by the axioms.
I have no idea what you mean by "nothing." That's not in the axioms. The empty set is not nothing. It's the empty set. A particular thing.
Quoting Metaphysician Undercover
Charming. You don't believe in abstract math, you don't believe in physics.
Quoting Metaphysician Undercover
You probably shouldn't bring up pi. You said the other day that pi is not a particular real number. That's a statement so monumentally ignorant that I either have to ignore it or stop responding to you altogether. So far I'm just trying to ignore it. Why you'd bring it up again, I don't know. You're just reminding me what a monumental waste of time this is.
Quoting Metaphysician Undercover
Every subset of the natural numbers is a set.
Quoting Metaphysician Undercover
I'm afraid "unity" is not mentioned in the axioms. You keep making things up. You are unable to focus on what's in the axioms. It's like someone trying to teach you chess and you say, "Well the knight must wear armor and save damsels," or "The knight must be "a man who served his sovereign or lord as a mounted soldier in armor." No no no no no. The knight in chess is exactly what the rules say the knight is. You don't get the concept of formal rules, fine. I doubt you're like this in real life, and you're quite tedious to regress to this infantile obfuscatory state here. I thought we'd moved a little past that, but apparently not.
Quoting Metaphysician Undercover
But I never said a set is a unity. I don't know what a unity is. It's not mentioned in the axioms.
Quoting Metaphysician Undercover
I agree that objection has been raised against set theory. It's not a point I'm interested in debating. Thoralf Skolem pointed out that the concept of set is far less coherent than people imagine. Many mathematicians and philosophers have made the point. For purposes of discussion, I'll even concede the point. But it's irrelevant. It doesn't diminish or change set theory, which is a particular formal system that need not have any referent or even be entirely sensible. It's just a list of formal symbols and the game is to derive their logical consequences.
If you don't want to play chess, that's fine. But for you to stand on a soapbox in the middle of town and rant and rail about chess, that's another thing entirely. You don't like set theory, you get no argument from me. I like set theory but I don't think others need to. But your vociferous objections to the reality of set theory are a waste of time. I don't make any claims it's real. It's just a formal system that some people find interesting, and that gained 20th century mindshare as the foundation of math. In fact set theory is all the more interesting lately, "now that it's been relieved of its ontological burden," as one set theorist put it.
ps -- @Meta let me sum this up. A couple of weeks ago I noticed that you are taking Frege's side in the great Frege-Hilbert controversy; namely, that you claim axioms must mean something or refer to something. Hilbert says no, that the theorems must be true of beer mugs and tables.
Since you feel that way, it's not something I can talk you out of. There is no right or wrong position. In real life Frege refused to "get" modern math and Hilbert stopped returning his letters. Likewise you don't want to get modern math. That's your right. But there is no point in your repeating these same talking points. The axioms of set theory are what they are. There are no "causes" or "reasons" nor "inherent order." I can't argue these points with you anymore.
We made a bit of progress when you started to at least acknowledge the reality of modern set theory. But if you don't want to build on that, I can't argue you out of your position and I wouldn't if I could.
pps --
Quoting Metaphysician Undercover
LOL.
:100:
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Hi.
So it seems to me a number is a "unity" and a set is not a noun but more like a verb. It's our action of containing a unity or many unities or unities and containers (verbs). I've been considering the "set of all sets that do not contain themselves" vs the "set of all sets the do contain themselves". This leads to what I see as Hilbert's position (contra Frege) of our rational power of humans to think of thinking of thinking of thinking and on to infinity. The set\verb would take precedence over the unity\number we place before our eyes as an object.
I always get into trouble with these philosophically loaded terms. Any number can be broken up into parts. 2 = 1 + 1, 1 = 1/2 + 1/2. So nothing in math is "indivisible." If anything at all is, it would be a pure Euclidean point, or a single real number representing a location on the number line. But what of it? Making a mystery or a big deal out of the idea that something is a "unity" doesn't speak to me; and it's one of the points where I do get in trouble in these philosophical discussions. A number is a number. It might be a real number, representing a signed distance on the number line. Or it could be a complex number, representing a rotation and stretching operator in the plane. Or a quaternion, used by game programmers as a nice formalism for rotating things in 3-space. I know the formalism of how numbers are represented in set theory; and I know that numbers aren't "really" sets; rather, they're abstract things that are pointed to by their various representations. What that means, I don't worry about too much. I've read a bit of the literature, I was reading up on structuralism the other day when that came up in one of the discussions on this site.
But I don't know what a unity is, or whether a number is one. I looked it up on Google, and it says that, "Unity, or oneness, is generally regarded as the attribute of a thing whereby it is undivided in itself and yet divided from others."
Well ok. But other than 0 and an individual point in space, I don't know what it means for any mathematical entity to be "undivided in itself." Actually only a point on a line has no parts. A point in space, say 3-space, is given by three spacial coordinates (x,y,z). And that's three things! I can take a point's projection onto the x, y, and z axes, to find that its "components" are x, y, and z, respectively. So even a point in space has components. I don't know what philosophers make of that.
All it all I can't agree or disagree that a number is a unity. I don't even know what that means.
Quoting Gregory
I see what you mean. You have an apple and an orange, and forming a set {apple, orange} is an act of gathering. It's quite mysterious and not entirely coherent, a point @Metaphysician Undercover has made and that I somewhat agree with. I don't know what it means to form a set out of individual objects. When pressed I can fall back on the formalism of the axiom of pairing, one consequence of which is the fact that if I have a mathematical object [math]X[/math], I'm allowed to form [math]\{X\}[/math], "the set containing" [math]X[/math]. I totally understand and accept that mathematically. Metaphysically, I don't know for sure that it's even a coherent concept.
That in fact is one of @Metaphysician Undercover's frequent points. What he doesn't understand is that I totally agree with him. Or at least I do for sake of discussion. It's not a hill I need to die on. I make no claim that set theory is coherent or sensible. Only that it's a formal system of rules that some people find interesting. I don't reify set theory or put it on a pedestal or make any claims about it. Like the novel Moby Dick. It's not a true story, but it's worthwhile nevertheless. It's based on a true incident, but only very loosely. If someone wants to tell me that set theory is incoherent, I don't object to that point of view. It doesn't matter. It's interesting on its own terms; and massively useful in formalizing most of modern math. What more can you ask of a formalism?
Which is to say that if set-collection is regarded by you as a verb, I do see your point. The act of gathering individuals into a mathematical set is a great act of abstraction that leads to many counterintuitive results. It's a powerful concept, even if not entirely coherent.
Quoting Gregory
Russell's paradox just shows that we can't form the set of all sets; and in fact that we can't form sets out of arbitrary predicates. Now that's very profound. Originally it was thought that if P is a predicate, then the collection of all the things that satisfy P form a set. That turns out to lead to a contradiction. Rather, a set is nothing more or less than exactly what the axioms of set theory say they are. Which for some philosophers is not a very satisfactory state of affairs. What I do know is that "high school sets," which are collections of similar or related objects, are nothing like actual mathematical sets. Actual mathematical sets are far stranger than that.
Quoting Gregory
I was trained in modern mathematical abstraction and have a hard time understanding Frege's point of view. It upsets some people (Frega, @Meta) that mathematical axioms don't necessarily "mean" anything or "refer" to anything. As Hilbert said, "“One must be able to say at all times
— instead of points, straight lines, and planes — tables, chairs, and beer mugs.” Whether he truly believed that, or was only retreating behind the formalist view because a realist mathematical stance is untenable, I don't know. Hilbert's formalist dreams were blown up by Gödel. There's a realm of mathematical truth that exists outside of anything we can capture with axioms.
And it should do, for classical set theory and real analysis are misleading and unrepresentative nonsense, unless cut down to the computationally meaningful content. Students who are taught those subjects aren't normally given the proviso that every result appealing to the axiom of choice is nonsensical, question-begging and of use only to pure mathematicians and historians.
I'm perfectly happy to stipulate so for purposes of discussion. After all, there are no infinite sets in physics, at least at the present time. So, what of it? The knight doesn't "really" move that way. Everybody knows that knights rescue damsels in distress, a decidedly sexist notion in our modern viewpoint. Therefore chess is misleading and unrepresentative nonsense. Nevertheless, millions of people enjoy playing the game. And millions more enjoy NOT playing the game. What I don't understand is standing on a soapbox railing against the game. If math is nonsense, do something else. Nobody's forcing you to do math, unless you're in school. And then your complaints are not really about math itself, but rather about math pedagogy. And I agree with you on that. When I'm in charge, a lot of state math curriculum boards are going straight to Gitmo.
Quoting sime
May well be so. I still think the way the knight moves is nonsensical too. What of it? You don't find me down at the park yelling at the chess players. Why is this a concern to you?
I might point out, though, that assuming the negation of the axiom of choice has consequences every bit as counterintuitive as assuming choice. Without choice you have a vector space that has no basis. An infinite set that changes cardinality if you remove a single element. An infinite set that's Dedekind-finite. You lose the Hahn-Banach theorem, of vital interest in functional analysis, which is the mathematical framework for quantum mechanics. The axiom of choice is even involved in political science via the Arrow impossibility theorem.
Besides, if you have a nation made up of states, can't you always choose a legislature? A legislature is a representative from each state. If there were infinitely many states, couldn't each state still choose a representative? The US Senate is formed by two applications of the axiom of choice. The House of Representatives is a choice set on the 435 Congressional districts. The axiom of choice is perfectly true intuitively. If you deny the axiom of choice, you are asserting that there's a political entity subdivided into states such that it's impossible to form a legislature. How would you justify that? It's patently false. If nothing else, each state could choose a representative by lot.
Thanks for sharing your wisdom on these types of threads
I'm not reverting back. Just because I understand better what I didn't understand as well before, doesn't mean that I am now bound to accept the principles which I now better understand.
Quoting fishfry
I suggest you look into the concept of infinite regress. The negative numbers are not an example of infinite regress.
Quoting fishfry
No, you said "set" has no definition, as a general term, and I went along with that. But I spent a long time explaining to you how a set must have some sort of definition to exist as a set. You seem to be ignoring what I wrote. Since you haven't seriously addressed the points I made, and you claim not to be interested, I won't continue.
Quoting Luke
Of course not, that's contradictory. According to the terms of the law of identity, two distinct things are not the same thing, so "two distinct but same things" is contradictory if we adhere to the definition of "same" provided by the law of identity.
Quoting Luke
Those are similar but different things, therefore not the same.
Quoting Luke
I still don't see your point, or the relevance.
An infinite regress in the real world would simply be past time encompassing the negative numbers as they move by the laws of physics. You might find such a past unsatisfactory without perfect unity undergirding it but that comes from your particular spirituality
This is Stanford Encyclopedia of Philosophy on infinite regress. "An infinite regress is a series of appropriately related elements with a first member but no last member, where each element leads to or generates the next in some sense."
Notice that there is a starting point, and this is why infinite regress is a logical problem, there is generally an assumption which requires something else for justification, and this requires something else etc.. Numbers in themselves, do not constitute an infinite regress because a number itself does not require a next number for justification. We may justify with the prior number, and finally the concept of "one", "unity", which is grounded in something other than number. So infinite regress in numbers is axiom dependent.
Notice the statement that "few have found this worrying". This is because, as fishfry demonstrates, "pure mathematicians" are wont to create axioms with total disregard for such logical problems which are entailed by those axioms. In other words, there are many issues which philosophers see as logical problems, but mathematicians ignore as irrelevant to mathematics. As pure mathematicians proceed in this way, the logical problems accumulate. This has created the divide between mathematics and philosophy which fishfry and I touched on in the other thread, in reference to the Hilbert-Frege disagreement.
You seemed to be reverting back to the Frege-Hilbert paradigm, which is a pointless discussion because there is no right or wrong, just a different worldview. I can't talk you out of yours nor would I if I could.
Quoting Metaphysician Undercover
In fact they are. They are often used philosophically as a model of infinite regress of causation. If you say cause -47 causes -46 which causes -45 etc., you have a model of causality in which (1) every effect has a direct cause, yet (2) there is no first cause. That's infinite regress.
ps -- I looked at the SEP article. That is utterly bizarre. An infinite regress goes backward without a beginning. Going forward without end like the Peano axioms is not an infinite regress. I refer to all the standard cosmological arguments, for example William Lane Craig's Kalam cosmological argument, where he argues against infinite regress going backwards. I have never seen infinite regress defined incorrectly as going forward as in this SEP article. The author made a mistake.
You might check the Wiki article on infinite regress, which is itself a little vague but at least correct. The SEP piece confuses induction with infinite regress. That's false. Induction always has a base case. Infinite regress fails to have a base case, that's what makes it an infinite regress.
Quoting Metaphysician Undercover
You are equivocating two senses of "definition." The word set has no definition in set theory. You can consult the axioms of Zermelo-Fraenkel set theory to verify this.
On the other hand, some sets do have definitions, or more accurately, specifications. For example the set of prime numbers, the set of even numbers, the set of counterexamples to Fermat's last theorem. That latter by the way is the empty set. See the axiom schema of specification to understand how SOME sets may indeed have a specification.
But MOST sets can't possibly have specifications, because there are more sets than specifications, a point I've made several times and that you prefer not to engage with. There are uncountably many sets and only countably many specifications. There simply aren't enough specifications to specify all the sets that there are. Most sets are simply collections of elements unrelated by any articulable property other than being collected into that set.
And a set can't ONLY be given by a specification, because then I'll give you the specification [math]x \notin x[/math] and get the Russell paradox.
It's a bit like numbers. There is no general definition of number; but there are specific definitions of the real numbers, the complex numbers, the p-adic numbers, the quaternions, and so forth.
Quoting Metaphysician Undercover
I carefully read everything you write. And either refute it or place it in context or give my own point of view. And then you come back with the same misunderstandings and ignore my refutations.
Quoting Metaphysician Undercover
I have addressed all of your points and either refuted them or placed them in context.
Quoting Metaphysician Undercover
I'm not interested in recapitulating the Frege-Hilbert dispute, since it's a matter of worldview and I could not change your mind because there is no right or wrong about the matter. And I'm in good company, because in the end Hilbert simply stopped responding to Frege's letters. If you reject abstract axiomatic systems, nobody can talk you out of that viewpoint.
Quoting Metaphysician Undercover
Thanks for the chat, then. All the best.
Quoting Gregory
You're welcome. Very much appreciated.
I agree. It's nonsense. Regress means going backward. I am more than familiar with these notions, as I investigate dynamical processes going forward as well as those going backward.
"Lastly, logical regress does not apply in mathematics" - OK, thank goodness.
Quoting jgill
Another example of the division between mathematics and philosophy. But the Wikipedia entry is consistent with the SEP.. You two just seem to twist around the concept, to portray infinite regress as a process that has an end, but without a start, when in reality the infinite regress is a logical process with a start, without an end.
Perhaps it is the idea of "forward" and "backward" which is confusing you. There is no forward and backward in logic, only one direction of procedure because to go backward may result in affirming the consequent which is illogical.
Quoting fishfry
This is what I've argued is incoherent, the assumption of an unspecified set, and you've done nothing to justify your claim that such a thing is coherent. I will not ask you to show me an unspecified set, because that would require that you specify it, making such a thing impossible for you. So I'll ask you in another way.
We agree that a set is an imaginary thing. But I think that to imagine something requires it do be specified in some way. That's the point I made with the distinction between the symbol, and the imaginary thing represented or 'specified' by the symbol. The symbol, or in the most basic form, an image, is a necessary requirement for an imaginary thing. Even within one's own mind, there is an image or symbol which is required as a representation of any imaginary thing. The thing imagined is known to be something other than the symbol which represents it. So, how do you propose that an imaginary thing (like a set), can exist without having a symbol which represents it, thereby specifying it in some way? Even to say "there are sets which are unspecified" is to specify them as the sets which are unspecified. Then what would support the designation of unspecified "sets" in plural? if all such sets are specified as "the unspecified", what distinguishes one from another as distinct sets? Haven't you actually just designated one set as "the unspecified sets"?
Ah, the good old daze. That didn't last long.
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
It's the distinction between two linear orders:
... < a4 < a3 < a2 < a1 < a0.
There's no first element but there is a last element. The earth (a1) rests on the back of turtle a1, which rests on the back of turtle a2, and so forth. It's turtles all the way down.
The SEP article reverses this:
a0 < a1 < a2 < a3 < ...
The earth (a0) has a turtle on it, a1; which has a turtle on its back, a2, and so forth. It's turtles all the way UP!
Now I recognize that in some sense these structures are "the same," in the sense that they just a mirror image of each other. Technically we would say that there is an order anti-isomorphism between them.
But the semantics are completely different. In the first model, there is no uncaused cause. William Lane Craig would argue (sophistically, but whatever) that this is impossible; that there MUST be a first cause, which is not only God, but is the Christian God. That's the argument.
So my contention is that the SEP article flips the direction of what an infinite regress is. And that's not necessarily a mathematical view. It's the philosophical view too. Turtles all the way down versus turtles all the way up.
Quoting Metaphysician Undercover
I will gladly show you an unspecified set, one of the classic cases. It's called the Vitali set. Consider a binary relation [math]\sim[/math] defined on pairs of real numbers by:
[math]x \sim y[/math] if [math]x - y \in \mathbb Q[/math]
That is, two real numbers [math]x[/math] and [math]y[/math] are related by [math]\sim[/math] if their difference is rational. For example [math]\pi + \frac{1}{2} \sim \pi[/math]. That's because their difference, 1/2, is rational.
You can verify that the relation [math]\sim[/math] is reflexive (every real number is related to itself, since the difference with itself is zero, which is rational); symmetric: if [math]x \sim y[/math] then [math]y \sim x[/math]; since the difference in one direction is just the negative of the difference in the other, so they're either both rational or neither are; and transitive, meaning that if [math]x \sim y[/math] and [math]y \sim z[/math] then [math]x \sim z[/math]. You should see if you can convince yourself that this is true.
A binary relation that is reflexive, symmetric, and transitive is called an equivalence relation. There is a basic theorem about equivalence relations, which is that they partition a given set into a collection of pairwise disjoint sets whose union is the original set.
So [math]\sim[/math] partitions the real numbers into a collection of pairwise disjoint subsets, called equivalence classes, such that every real number is in exactly one subset. By the axiom of choice there exists a set, generally called [math]V[/math] in honor of Giuseppe Vitali, who discovered it, such that [math]V[/math] contains exactly one member, or representative, of each equivalence class.
You can tell me NOTHING about the elements of [math]V[/math]. Given a particular real number like 1/2 or pi, you can't tell me whether that number is in [math]V[/math] or not. The ONLY thing you know for sure is that if 1/2 is in [math]V[/math], then no other rational number can be in [math]V[/math]. Other than that, you know nothing about the elements of [math]V[/math], nor do those elements have anything at all in common, other than their membership in [math]V[/math].
Is this example important? Yes, it's part of the foundation of modern probability theory.
[math]V[/math] is the classic example of a nonconstructive set. I don't expect you to regard this as particularly intuitive. It's an example typically shown to first-year grad students in math. It takes a while to get your mind around it. But "whether you like it or not," as Gavin Newsom said about gay marriage, [math]V[/math] is a perfectly legitimate set in ZFC, and actually turns out to be of theoretical importance.
You have now seen the classic example of a nonconstructive set.
Quoting Metaphysician Undercover
You're wrong. I just demonstrated a specific example, one that is not only famous in theoretical mathematics, but that is also important in every field that depends on infinitary probability theory such as statistics, actuarial science, and data science.
I know you have an intuition. Your intuition is wrong. One of the things studying math does, is refine your intuitions.
Quoting Metaphysician Undercover
What distinguishes one set from another as distinct sets? Their elements, as expressed by the axiom of extensionality, as I've explained to you at least a dozen times in the past year.
It's like being out in a field picking daisies. You pick this daisy, you pick that daisy. When you're done, you have a basket full of daisies. Must they have some particular property in common for you to have picked them? No, you picked them randomly. The only thing they have in common is that you picked them. For no reason at all. It's just like this week's winning lottery numbers. They have nothing in common other than that they were picked randomly. Once you start thinking about it that way, you'll find many such examples in daily life of perfectly random sets. A bunch of people check into a hotel.What do they have in common that distinguishes them from all other human beings? Nothing at all, except that they all checked into the hotel.
If you know nothing else about mathematical sets, know this: A set is entirely characterized by its elements.
As Judge Reinhold said to Sean Penn in Fast Times at Ridgemont High: Learn it. Know it. Live it.
Awfully good catch, thank you. Especially since the footnotes don't appear on the article page and must be clicked on to see them at all. I commend your attention to detail in clicking on the footnotes.
I am of two minds on this. On the one hand yes, the footnote is correct and there is fundamentally no difference mathematically. The two interpretations are order anti-isomorphic, just mirror images of each other.
Still, to me the semantics are profoundly different. I guess I have to accept that in the end the difference is not important. Still @Metaphysician Undercover must also agree that when he says that @jgill and I have infinite regress wrong, he's incorrect about that too. If both interpretations are the same, everyone's right.
Thanks for clicking on that footnote!
ps -- I just can't agree with SEP, period. The article is wrong and I'm right. My basis for this belief is the concept of well-foundedness, which is essential to set theory and is encoded as the axiom of foundation.
In set theory it is legal to have infinitely upward membership chains [math]x_0 \in x_1 \in x_2 \in \dots[/math]
In fact this is not only legal, it's standard, as exemplified by the finite von Neuman ordinals [math]0 \in 1 \in 2 \in \dots[/math].
Whereas it is expressly forbidden to have infinitely downward membership chains [math]\dots \in x_2 \in x_1 \in x_0[/math].
Even though the two conditions are mirror-images of each other, set theorists strongly distinguish between the two; considering one situation normal and the other illegal. I rest my case, and will probably drop the author a note with my two cents.
You are specifying "the real numbers". How is this not a specification?
Quoting fishfry
Actually, you're wrong, your set is clearly a specified set.
Quoting fishfry
This is not true, you have already said something else about the set, the elements are real numbers.
Quoting fishfry
I'll agree with Tones, the two ways are just different ways of looking at the same thing. That's why I said the Wikipedia article is consistent with the SEP. I do believe there are metaphysical consequences though, which result from the different ways, or perhaps they are not consequences, but the metaphysical cause of the difference in ways. The principal consequence, or cause (whichever it may be), is the way that we view the ontological status of contingency.
The real numbers include some numbers that are in [math]V[/math] and many that aren't. In what way does that specify [math]V[/math]? That's like saying I can specify the people registered at a hotel this weekend as the human race. Of course everyone at the hotel is human, but humanity includes many people who are not registered at the hotel.
Quoting Metaphysician Undercover
How so? I gave an existence proof. In no way did I tell you how to determine which real numbers are in it and which aren't. Can you explain your thought process?
Quoting Metaphysician Undercover
And the people at the hotel are humans. As are all the people not at the hotel. If that's all you mean by specification, that all I have to do is name some arbitrary superset of the set in question, then every set has a specification. If that's what you meant, I'll grant you your point. But it doesn't seem too helpful. It doesn't tell me how to distinguish members of a set from non members.
It's like an exclusive club that only allows in certain people. You run the club. You hire a doorman. He asks, "How can I tell who's a member or not?" And you say, "Oh just let everybody in." What kind of specification is that?
Quoting Metaphysician Undercover
I agree that there is a point of view by which it makes no difference whether you define infinite regress as going forward or backward. And a sense in which it makes a huge difference, such as well-foundedness. It can be argued either way.
Here's a counterpart of this idea in mathematics. (1) is called left composition or outer composition, and (2) is called right composition or inner composition. (1) and (2) are very different ideas. But to compound difficulties in language or notation, (2) is usually numerically evaluated using backward recursion, which is very efficient:
(1) [math]{{G}_{n}}(z)={{g}_{n}}\circ {{g}_{n-1}}\circ \cdots \circ {{g}_{1}}(z),\text{ }G(z)=\underset{n\to \infty }{\mathop{\lim }}\,{{G}_{n}}(z)[/math]
(2) [math]{{F}_{n}}(z)={{f}_{1}}\circ {{f}_{2}}\circ \cdots \circ {{f}_{n-1}}\circ {{f}_{n}}(z),\text{ }F(z)=\underset{n\to \infty }{\mathop{\lim }}\,{{F}_{n}}(z)[/math]
The point is that basing your mathematical "principles" on empiricism or reality demonstrably leads to absurdity, including your rejection of fractions, negative numbers, imaginary numbers, infinity, circles, probabilities, possible set orderings, and potentially all mathematics. Instead of coming to realise that this indicates a serious problem with your principles and position, you continue in your delusion that you possess a superior understanding of mathematics.
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That's right, to specify that they are real numbers is to specify, just like to specify that the guests at the hotel are human beings is to specify. The fact that a specification is vague, incomplete, or imperfect does not negate the fact that it is a specification.
Quoting fishfry
I told you how so. You've specified that the set contains real numbers. You are the one who explained to me, that 'set" is logically prior to "number", and that not all sets have numbers as elements. This means that "set" is the more general term. How can you now deny that to indicate that a particular set consists of some real numbers, is not an act of specifying?
Quoting fishfry
Good, you now accept that every set has a specification. Do you also agree now that this type of specification, which "doesn't tell me how to distinguish members of a set from non members", is simply a bad form of specification?
Anyway, let's go back to the point which raised this issue. You said the following, which i said was contradictory:
Quoting fishfry
Do you now see, and agree, that since a set must be specified in some way, then the elements must be "the same" in some way, according to that specification, therefore it's really not true to say that "the elements of a set need not be "the same" in any meaningful way." So we can get rid of that appearance of contradiction by stating the truth, that the elements of a set must be the same in some meaningful way. To randomly name objects is not to list the members of a set, because a set requires a specification.
What I am trying to get at, is the nature of a "set" You say that there is no definition of "set", but it has meaning given by usage. Now I see inconsistency in your usage, so I want to find out what you really think a set is. Consider the following.
Quoting fishfry
Since we now see that a set must have a specification, do you see how the above quote is inconsistent with that principle? Since a set must have a specification, a set is itself an "articulable category or class of thought". And, it is not the "being gathered into a set" which constitutes the relations they have with one another, it is the specification itself, which constitutes the relations. So if you specify a set containing the number five, the tuna sandwich you had for lunch, and the Mormon tabernacle choir, this specification constitutes relations between these things. That's what putting them into a set does, it constructs such relations.
Now here's the difficult part. Do you agree that there are two distinct types of sets, one type in which the specification is based in real, observed similarities, a set which is based on description, and another type of set which is based in imaginary specifications, a set produced as a creative act? Do you acknowledge that these two types of sets are fundamentally different?
So are you agreeing that mathematical infinity has neither philosophical nor scientific relevance and that everyone knows this, or am i right to stand on a soap box and point out the idiocies and misunderstandings that ZFC seems to encourage?
Quoting fishfry
Obviously, a denial of AC doesn't amount to an assertion of ~AC, given that things are generally undecidable, but i see no counter-intuitive examples in what you present. In fact, many examples you raise should be constructively intuitive if we recall that construction can proceed either bottom-up from the assumptions of elements into equivalence classes, or vice versa, so an inability to locate a basis in a vector space using top-down construction seems reasonable.
As for the sciences, AC is meaningless and inapplicable when it comes to the propositional content. At best, AC serves a crude notation for referring to undefined sets of unbounded size, but ZFC is a terribly crude means of doing this, because it only recognises completely defined sets and completely undefined sets without any shade of grey in the middle as is required to represent potential infinity.
QM has also been reinterpreted in toposes and monoidal categories in which all non-constructive physics propositions have been removed, which demonstrates that non-constructive analysis is dying and going to be rapidly replaced by constructive analysis, to the consternation of inappropriately trained mathematicians who resent not knowing constructive analysis.
Quoting fishfry
Obviously, the axiom of choice isn't used in the finite case. In the infinite case, the sets of states needs to be declared as being Kuratowski infinite in order to say that the elements of the set are never completely defined, and so a forteriori the size of the set cannot be defined in terms of it's finite subsets.
Secondly, the set should be declared as Dedekind finite, in order to say that the set is an observable collection of elements and not a function (because only functions can be dedekind-infinite).
So, yes, you can choose as many representatives as you wish without implying a nonsensical completed collection of legislatures that are a proper subset of themselves, but formalisation of these sets isn't possible in ZFC, because AC and it's weaker cousin, the axiom of countable choice, forces equivalence of Kuratowski finiteness and Dedekind finiteness.
If one doesn't wander into transfinite math, Choice is not necessarily required. A finite dimensional vector space doesn't require it for a basis. And if one works with a normed linear space that is separable, the Hahn-Banach theorem doesn't require it either. However, much modern math goes beyond these restrictions and requires transfinite results.
Quoting sime
When I write mathematical analysis programs for my computer I am certainly involved in constructive math, but pen on paper, not necessarily. I suspect constructive analysis will not overwhelm the math community, unless you know something I don't - which is possible. :cool:
Not entirely, but I'm not disagreeing either. Infinitary set theory has been used in the 20th century as the foundation of math, and math is the language of physics. So if we say that infinitary math is unrealistic or fiction or nonsense or false (any one of which I'd agree with for sake of discussion), we still have to explain the "unreasonable effectiveness" of math. So infinitary math is of great philosophical importance. And surely there are a whole lot of professional philosophers of math who do discuss and care about infinitary math. So infinitary math is both philosophically and scientifically relevant, even if we can plausibly argue that it's fiction. After all, as I'm fond of pointing out, Moby Dick is a work of fiction, but it still teaches us to avoid following our obsessions to our doom. Even fiction can be useful i the real world.
Quoting sime
I don't think everyone sees this issue the same way. There are a lot of different opinions, even different learned opinions.
Quoting sime
That's always a personal choice. I may dislike the designated hitter rule in baseball, but I don't go out and rant about it in public. One chooses one's battles. There are those who dislike contemporary mask mandates, but wear their mask as required by local laws. There are others who go into grocery stores and confront the hapless clerks. Each of us has many opinions, but we still have to choose which hills to die on and which soapboxes to stand on.
Quoting sime
I disagree with you about this. The way the knight moves in chess is a fiction, it's an arbitrary rule of a formal game. Does it produce idiocies and misunderstandings? No, it's just a rule of the game. I've seen prominent set theorists admit that they don't know if set theory is true or meaningful. Set theory is the study of certain formal structures. People do it because they find it interesting. Others are Platonists and believe they're seeking some higher truth.
I'm not sure what idiocies and misunderstandings you mean. If you don't like the way the knight moves, don't play the game. Or play some alternate variant of chess. There are many alternate variants of set theory. And billions of people live perfectly happy lives without ever knowing or caring about set theory.
So you're not "wrong," pe se, in disliking or objecting to infinitary math. But if in addition to that you have a strong emotional aversion to it, that's ... well, it's a personal issue. You might introspect as to why. Maybe you had a screechy math teacher in third grade. A lot of math anger started that way.
Quoting sime
Ok. Still, absent choice, infinite sets are badly behaved. There's a vector space without a basis, a surjection without a right inverse (or section), a commutative ring with unity with no maximal ideal. These things are very inconvenient in math.
And that's the biggest reason for adopting AC versus rejecting it. Convenience. The reasons for adopting or rejecting axioms are pragmatic. We are not asserting any kind of absolute truth. We're only choosing the axioms that make it convenient to do math. Maddy explains all this in her classic articles, Believing the Axioms, I and II. We want expansive rather than restrictive axioms, and so forth.
Quoting sime
If you don't find a vector space without a basis, a surjection without a right inverse, an infinite set that changes cardinality when you remove one element, etc., counterintuitive, then we see that differently. In the end, AC makes infinite sets well-behaved. For example without AC there are the Alephs, and then there are many infinite cardinalities that aren't Alephs. With AC, all the cardinals are Alephs. There's no "absolute truth" in that, just convenience.
I think if you regard AC as a pragmatic choice, it's easier to understand. It relieves you of needing to stand on a soapbox. Mathematicians are choosing convenience and a more orderly and expansive set-theoretic universe. What need is there to stand on a soapbox against someone's pragmatic choices?
Quoting sime
I suppose I can concede your point that AC is not strictly necessary for the sciences. Nor is the knight move necessary to cook a lasagna.
Quoting sime
ZFC is not even a 100 years old. In its present form it dates from Zermelo's 1922 axiomatization. And Mrs. Zermelo was pro choice. (/joke). There's no telling how these matters will be seen in another hundred years. As Max Planck said, science proceeds one funeral at a time. Meaning that the old guard die off and a new generation grows up accepting the new ideas.
Quoting sime
Why can't a nation in some alternate universe have infinitely many states, and choose a legislature? My argument here is that although AC is independent of ZF, it's nevertheless intuitively true. Even the ld joke admits that. AC is definitely true, the well-ordering theorem is definitely false, and Zorn's lemma, who knows! The joke being that they're all logically equivalent.
Quoting sime
You lost me there. Absent AC there is a set that is infinite (not bijective with any natural number) yet Dedekind-finite (no proper subset is bijective with the entire set). I don't know what you mean here.
Quoting sime
I don't think you're engaging with my point. Why can't the uncountably many provinces of the planet Zork choose themselves a legislature? Why on earth can't we form a set consistent of one element from each of a collection of nonempty sets? It's intuitively true that we can, even if AC is not provable from ZF.
Besides, Gödel showed that AC is true in his constructible universe in which, as I understand it, all ordinary mathematics takes place anyway. So there is a perfectly good model of math in which AC is true. You can't stand on a soapbox and deny that.
But if you prefer to do math without AC, or if you are a constructivist, or a finitist or ultrafinitist, you're in good company. I can't argue you out of your preferences. I can only question your soapbox emotions. After all most serious constructive mathematicians still use some version of AC, because without it it's more difficult to do math.
Only if you change what a specification is. In set theory, a specification is a predicate, a statement that can be true or false of a given item. The items for which the predicate is true, go into some set.
By saying, "Oh, they're all real numbers" when the set in question only contains SOME of the real numbers, does not specify the set.
But your definition of specification is not even valid by the everyday English meaning of the word. If you go into a store to buy a computer and you ask the salesman for the specifications of a particular model you have in mind, and they say, "Oh, it's a material object made of atoms," you'd walk out of the store. He has told you the truth, but has not given the specifications of the computer.
Likewise if you are a manufacturer and you subcontract out a particular part, asking to have it made to a particular specification, and they send you back a lump of metal totally unlike what you asked for, saying, "Well it's a physical object made of atoms, what more to you want," you'd sue them for breach of contract.
Quoting Metaphysician Undercover
You have not specified which real numbers are in the set and which aren't. And if you really believed what you are saying, you would have told me long ago that every set is either empty or contains other sets; and THAT is a specification. Is that what you claim? Every set is fully specified by saying its elements are other sets? That's nonsense. That's not what a set specification is.
A set specification consists of two things: One, an existing set; and two, a predicate saying which elements of the given set are members of your specified set. You haven't done that. You've made a trivial and sophistic point, saying that every set of real numbers is "specified" because it contains real numbers. That's nonsense. It's childish.
Quoting Metaphysician Undercover
Only by completely changing both the mathematical AND the everyday meaning of specification.
Quoting Metaphysician Undercover
Abraham Lincoln used to ask, If you call a tail a leg, how many legs does a dog have? And he answered: Four. Calling a tail a leg doesn't make it a leg.
Likewise, calling your vague and sophistic characterization of a set a specification does not make it a specification. Of course it could be argued that logically IF I call a tail a leg, then a dog has five legs. In THAT sense, you have made your point. Which is to say: You haven't made your point.
Quoting Metaphysician Undercover
Not in the least. Of course in pure set theory (ie set theory without urelements), the elements of every nonempty set are other sets. So it's true that the elements of every set are sets, and they have that in common. But that is not a property that distinguishes any given set from all other sets, so it's not a specification. If you want to call it a specification that's your right, just as you can call a tail a leg and say that a dog has five legs. If you can get anyone to take you seriously.
Quoting Metaphysician Undercover
You haven't specified a set by pointing out that all its elements are sets. Because that doesn't distinguish that set from any other set. But like I say, you can hang on to your childish sophistry or you can hang on to your credibility. You might as well hang on to the former, having long ago lost the latter.
Quoting Metaphysician Undercover
Again -- AGAIN -- you are equivocating between the definition of a set in set theory, of which there is none -- just check the axioms please -- and the fact that some sets are given by specifications, where a specification consists of an already-existing set and a predicate. See the axiom schema of specification for an explication of this point. It's called a schema because it actually consists of infinitely many axioms, one for each predicate.
Quoting Metaphysician Undercover
The Vitali set has no specification. It's true that all its elements are real numbers; and in fact all its elements are also sets, since in set theory, real numbers are modeled as sets. But that's as I said a childish and sophistic point, which you can hold to only at the loss off your own credibility.
Quoting Metaphysician Undercover
No, I see you making an astonishingly childish and sophistic point, calling a tail a leg and saying a dog has five legs.
Quoting Metaphysician Undercover
Most sets have no specifications. You've lost all credibility with me at this point. Well to be fair, you lost all credibility last week when you denied that pi is a particular real number. I have no idea what you might be thinking, but whatever it is, it's wrong.
Quoting Metaphysician Undercover
Sure, trivially. But not meaningfully, since it's circular. How do I tell which real numbers are in the Vitali set? Well, they're in the Vitali set. Not helpful.
Quoting Metaphysician Undercover
If it makes you happy to hold to this line of argument, I would not take that away from you.
Quoting Metaphysician Undercover
Oh this should be good.
Quoting Metaphysician Undercover
Only to the extent that your first category is empty. You ask if there's a set based on "observed" similarities. I have never observed a set. I've been to math grad school and never observed a set. Set's are entirely abstract objects and are not observable in the sense of physics nor in the sense of everyday English. You cannot observe a set.
And "real?" What on earth do you mean by that?
FWIW I will grant that you might have meant to ask: Are there some sets given by the axiom of specification; and others that are not, and that are essentially nonconstructive? In which case yes, the set of even natural numbers is an example of the former, and the Vitali set the latter. But neither set is particular "real" or "observable." I have never observed an even number. I've seen four apples. And after they downgraded poor old Pluto, I then knew about the eight planets. But four? Or eight? I've never observed them. They're both abstract entities.
Quoting Metaphysician Undercover
Sure. All the sets that there are, are in the latter category; and none are in the former. No sets are based on anything "real" or "observed," and all sets are "based in imaginary specifications, a set produced as a creative act." Those are the only types of sets there are.
Just as there are two types of elephants: those that fly, and those that don't fly. That is a true statement. It's just that all the elephants are in the latter category and none in the former.
But being charitable and assuming you meant to ask about constructive and nonconstructive sets, sure. Constructivist mathematicians make the distinction all the time. Just ask @sime, who rejects the axiom of choice. He does not believe in the Vitali set (I assume, though we've never directly discussed it), and believes all sets are the output of some algorithm or deterministic process. It's a fair distinction.
Nice to have an actual mathematician around here!
OED: specify, "to name or mention". Clearly the set you called "V" is not unspecified, and it's you who wants to change the meaning "specify" to suit your (undisclosed) purpose. Sorry fishfry, but you appear to be just making stuff up now, to avoid the issues.
Whatever, man.
"suit your (undisclosed) purpose" -- what does that mean? I'm using specification as in the axiom schema of specification. We are talking about set theory after all. If we're talking about baseball, a "fly" is a ball that's hit by the batter and remains in the air without touching the ground. It's not a winged insect.
What "undisclosed purpose" would I have? The corruption of the youth? What are you talking about?
Nevermind, I don't want to know.
If as you agree, all sets in standard set theory are composed of nothing but other sets; and that therefore every nonempty set whatsoever can be said to have elements that are sets; then isn't the fact that the elements of any set have in common the fact that they are sets, a rather trivial point? Can you see that this is not a helpful criterion to specify which elements are in the set and which are not? It's the example I gave of the doorman at a highly exclusive club who's told to just let everyone in. To be a valid specification, you have to tell the doorman which people to let in, and which to not let in.
But like I said, I am perfectly willing to accept your personal definition of the word; at the cost of no longer being able to take you even slightly seriously; since you're making such an unserious point.
Quoting TonesInDeepFreeze
Not sure I follow. You mention Z a lot but that's a pretty obscure system unless one is a specialist. AC, ZL, and WO are surely equivalent in ZF. In what way would you say they're not equivalent?
But one left on the ground while math, like Buzz Lightyear, has gone "To infinity and beyond ! "
Z |- AC <-> ZL & ZL <-> WO & AC <-> WO
But it is not the case that
|- AC <-> ZL & ZL <-> WO & AC <-> WO
Is Z supposed to be Zermelo set theory? In Wiki they call it [math]Z^-[/math], is that the same thing?
https://en.wikipedia.org/wiki/Zermelo_set_theory
Why do you work in Z so much, why do you care, why exactly should I care? I'm not familiar with it. Perhaps you can supply some context please.
Quoting TonesInDeepFreeze
Why is that last line not true? What exactly do you mean? Are you just saying it's not a tautology? Can you give a model in which it's false? In my experience, the claim that "AC, WO, and ZL are equivalent" is totally noncontroversial and nobody (but you) would look twice at it. Why do you take exception?
What I'm looking for is context and understanding of your thought process.
extensionality
schema of separation
pairing
union
power
infinity
regularity
It is uncontroversial that AC, ZL, and WO are equivalent in the sense that in Z (perforce ZF) we can prove one from the other. But it is not the case that they are logically equivalent. To be logically equivalent they would have to be provable one from the other using only the pure predicate calculus.
Absent AC, it is undecided whether there is such a set.
Yes ok I meant assuming the negation. But if you don't specify one or the other, you're right.
I don't see that as a trivial point, because not only is "set" undefined, but also "element" is undefined. So we have a vicious circle which makes it impossible to understand what type of thing a set is supposed to be, and what type of thing an element is supposed to be. What is a set? It's something composed of elements. What is an element? It's a set.
Under this description, a particular set is identified by its elements, not by a specification, definition, or description. Do you see what I mean? Under your description, any particular set cannot be identified by the predicates which are assigned to the elements, because it is not required that there be any assigned predicates.. But there still might be such an identified set. So a set must be identified by reference to its members. This is why, under this description of sets, the empty set is logically incoherent. A proposed empty set has no members, and therefore cannot be identified.
If, on the other hand, a set is identified by it's specification, definition, or description, (which you deny that it is), then there could be a definition, specification, or predication which nothing matches, and therefore an empty set.
Hopefully you can see that the two, identifying a set by its elements, and identifying a set by its predications, are incompatible, because one allows for an empty set, and the other does not. So as much as "set" may have no formal definition, we cannot confuse or conflate these two distinct ways of using "set" without the probability of creating logical incoherency.
By saying that "set" has no definition, we might be saying that there is nothing logically prior to "set", that we cannot place the thing referred to by the word into a category. But if you make a designation like "there is an empty set", then this use places sets into a particular category. And if you say that a set might have no specification, this use places sets into an opposing category. If you use both, you have logical incoherency.
Therefore it is quite clear to me, that the question of whether a set is identified by reference to its elements, or identified by reference to its specification, is a non-trivial matter because we cannot use "set" to refer to both these types of things without logical incoherency.
What's trivial is saying that the Vitali set is "specified" because all its elements are real numbers. That's like saying the guests at a particular hotel this weekend are specified because they're all human. It's perfectly true, but it tells you nothing about the guests at the hotel. That's why your point is trivial.
Quoting Metaphysician Undercover
If [math]x \in y[/math] is true, then If [math]x[/math] is an element of If [math]y[/math]. That's perfectly well defined in terms of If [math]\in[/math], which is an undefined primitive. I've referred you many times to the axioms of Zermelo-Fraenkel set theory, or ZF, which if you'd read the page, might answer many of your questions.
It's also sometimes called ZFC in honor of Mrs. Zermelo, who was pro choice.
Quoting Metaphysician Undercover
On the one hand you're the only one who doesn't understand this. On the other hand, Skolem argued that the notion of set is too vague to be useful as a foundation for mathematics, and he was one of the greatest of the early set theorists. So you're not wrong. It would be better if you had more sophisticated arguments, because it would then be more interesting and fun to engage with you.
Quoting Metaphysician Undercover
If [math]x \in y[/math] then the thing on the left is an element of the set on the right. In pure set theory the thing on the left is also a set; and in set theories with urelements, the thing on the left might not be. Likewise in applications the thing on the left might be something else such as a voter in social choice theory, or a rational actor in an economic theory, etc.
Quoting Metaphysician Undercover
The axioms don't tell us. A set is characterized by its behavior under the axioms. Even you've agreed to that previously.
Quoting Metaphysician Undercover
So what? You want an explicit definition, but in set theory there are no such definitions. Read the axioms. You're just recapitulating Frege's complaints to Hilbert, but I can't argue the point because there's no right or wrong to the matter. You don't like it, that's your right.
Quoting Metaphysician Undercover
LOL. Yes, that's the axiom of extensionality, which I've been explaining to you for at least two years. Glad you finally got it. A set is entirely characterized by its elements.
Quoting Metaphysician Undercover
Some sets are specified by predicates, such as the set of all natural numbers that are prime. Some sets aren't specified by predicates, such as the Vitali set.
And as I've noted, constructivist mathematicians, neo-intuitionists, finitists, and ultrafinitists only believe in sets that can be constructed by an algorithm or explicit procedure. So this is a fair debate in the philosophy of math.
Quoting Metaphysician Undercover
Like the set of natural numbers that are prime. That's a set given by the axiom schema of specification.
Quoting Metaphysician Undercover
"By reference?" No. The Vitali set is characterized by its members, but I can't explicitly refer to them because I don't know what they are. It's a little like knowing that there are a billion people in China, even though I don't know them all by name.
Quoting Metaphysician Undercover
On the contrary. Since everything is equal to itself, the empty set is defined as [math]\{x : x \neq x\}[/math]. I rather thought you'd appreciate that, since you like the law of identity. The empty set is in fact the extension of a particular predicate.
Quoting Metaphysician Undercover
Exactly. The empty set is the extension of the predicate [math]x \neq x[/math]. Or if you like, it's the extension of the predicate "x is a purple flying elephant." Amounts to the same thing.
Quoting Metaphysician Undercover
Since the empty set is the extension of a particular predicate, your point is incoorect.
Quoting Metaphysician Undercover
You're wrong, since the empty set is the extension of a predicate.
Quoting Metaphysician Undercover
In set theory that's true. Although in the formal theory, the axioms and the inference rules of first-order predicate logic are taken as logically prior.
Quoting Metaphysician Undercover
Well it's in the category of sets. Which is formally true in the category of sets, and is also true in the sense of "category" that you're using. A set is a set.
Quoting Metaphysician Undercover
I don't see why. Every set is fully characterized by its elements. Specification is one particular way of identifying or constructing or proving the existence of particular sets. Other ways are union, intersection, and so forth. It's all given in the axioms, which I've given you the link to many times and which I'm sure you've never even bothered to glance at. You know the most credible way to make an argument is to take the trouble to become familiar with the thing you're arguing against. You won't do that. So you make very trivial and sophistic arguments.
Quoting Metaphysician Undercover
I don't see why. Now do I agree that constructive mathematicians do make this sharp distinction between sets that can be constructed by an algorithm or mechanical procedure (there are many sub-flavors of this idea) versus mathematicians who believe in nonconstructive sets such as the Vitali set. @sime, for example, is one who makes this distinction. If you wish to make an argument along these lines, that would be an interesting conversation. But I don't think that's what you're doing. I don't know what you're doing. i don't know what your point is.
tl;dr: There are many sophisticated arguments against set theory. Therefore you're not shocking me by questioning set theory. But your arguments are from ignorance rather than knowledge. So your arguments are not interesting.
ps -- Am I being to harsh to your ideas? I can't really follow your logic. The axioms are very clear. There's no circularity or infinite regress involved. There's no formal definition of set in the axioms; but some sets are given by specifications and others not. They're not "defined" by the specifications, rather their existence is given by the the axiom of specification. Some other sets are given by various other axioms. Replacement is one of the more subtle ones. Perhaps it's helpful to think of the axioms as a toolkit for determining which sets can exist. In that respect, specification isn't all that special. As to what it all means, perhaps it means nothing at all. That's not a problem, neither does chess.
It's not trivial, because it's a demonstration of what "specified" means. If you specify that the guests are all human, then clearly that is a specification. If you do not appreciate that specification because it does not provide you with the information you desire, then the specification is faulty in your eyes. But it's false to say that just because you think the specification is faulty, then there is no specification. There is a specification, but it is just not adequate for you. That is simply the nature of specification, it comes in all different degrees of adequacy, depending on what is required for the purpose. But an inadequate specification, for a particular purpose, is in no way a total lack of specification.
Quoting fishfry
Do you see then, that if "A set is entirely characterized by its elements", then a so-called empty set is not possible? If there are no elements, under that condition, then there is no set. A set is characterized by its elements. There are no elements. Therefore there is no set. If we adhere to this premise, "the set is entirely characterized by its elements", then when there is no elements there is no set.
Quoting fishfry
This is logically inconsistent with "a set is entirely characterized by its elements", as I explained in the last post. Either a set is characterized by its elements, or it is characterized by its specified predicates, but to allow both creates the incoherency which I referred to. One allows for an empty set, the other does not.
Quoting fishfry
We've been through this already. You clearly have referred to the members of the Vitali set. You've said that they are all real numbers. Why do you believe that this is not a reference to the members of the set? You can say "all the people in China", and you are clearly referring to the people in China, but to refer to a group does not require that you specify each one individually.
This seems to be where you and I are having our little problem of misunderstanding between us. It involves the difference between referring to a group, and referring to individual. I believe that when you specify a group, "all the guests at the hotel" for example, you make this specification without the need of reference to any particular individuals. You simply reference the group, and there is no necessity to reference any particular individuals. In fact, there might not be any individuals in the group (empty set). You seem to think that to specify a group, requires identifying each individual in that group.
This is the two distinct, and logically inconsistent ways of using "set" which I'm telling you about. We can use "set" to refer to a group of individuals, each one identified, and named as a member of that set (John, Jim, and Jack are the members of this set), or we can use "set" to refer simply to an identified group, "all the people in China". Do you see the logical inconsistency between these two uses, which I am pointing out to you? In the first case, if there are no identified, and named individuals, there is no set. Therefore in this usage there cannot be an empty set. But in the second case, we could name the group something like "all the people on the moon", and this might be an empty set.
Quoting fishfry
I must say, I really do not understand your notation of the empty set. Could you explain?
Quoting fishfry
This doesn't help me.
Quoting fishfry
Actually you don't seem to be getting my point. The point is that if a set is characterized by its predicates, then an empty set is possible, so I have no problem with "the empty set is the extension of a particular predicate". Where I have a problem is if you now turn around and say that a set is characterized by its elements, because this would be an inconsistency in your use of "set", as explained above. A set characterized by its elements cannot be an empty set, because if there is no elements there is no set. Do you apprehend the difference between "empty set" and "no set"?
Quoting fishfry
Perhaps it's a bit clearer now?
I've already agreed numerous times that if you insist on your own definition, you're right. You can't convince me that your way of looking at it isn't trivial. I originally understood you to be claiming that the members of every set had something in common that distinguished them of all the non-members. That's an interesting statement, and you'd find many constructivists who agree with you (to the extent that I understand the mindset of the constructivists).
I was surprised -- shocked, in fact -- to discover that you didn't mean that at all, but only meant that you could find some superset that contained the potential candidates for our set. That's a much weaker criterion, I hope you at least agree to that.
So yes, all the guests are human, but that hardly helps the security guard to know who to let in and who to keep out. And every element of the set of primes is a natural number, but that doesn't tell me what's a prime and what's not. It's an incredibly weak criterion. I'm happy to let you have it, but it's trivial because it's such a weak criterion as to be utterly useless in determining which elements are in a given set.
Quoting Metaphysician Undercover
First, the empty set is the unique set that has no elements at all. It's characterized by not having any elements.
But for a more precise answer, we have to look at the actual, exact formal statement of the axiom of extensionality. The natural language version, "A set is entirely characterized by its elements," is just an approximation to what the axiom actually says. The problem is that you don't relate to symbolic reasoning at all. That said, and for the record, I'll walk you through what the axiom of extensionality actually says; and for reference, you can see the Wiki link.
Axiom of extenionality: [math]\forall A \forall B (X \in A \iff X \in B) \implies A = B[/math]
We unpack this as follows. It says that
For all sets [math]A[/math] and [math]B[/math]:
If it happens to be the case that for all sets [math]X[/math], [math]X \in A[/math] if and only if [math]X \in B[/math];
Then [math]A = B[/math].
This says in effect that if two sets have exactly the same elements, they're the same set. But the way it's written, it also includes the case of a set with no elements at all. If you have two sets such that they have no elements, they're the same set; namely the empty set.
Now I know this isn't your cup of tea. And that's ok. All you need to know about this is that when you drill down into the technical details of what the axiom of extensionality actually says, the case of the empty set is included.
A more general point is that you want to criticize set theory based on vague natural language descriptions rather than grappling with the actual formal symbology. But in the end, we all have to roll up our sleeves and grapple with the symbology.
Another point is that everyone has trouble with vacuous arguments and empty set arguments. If 2 + 2 = 5 then I am the Pope. Students have a hard time seeing that that's true. The empty set is the set of all purple flying elephants. A set is entirely characterized by its elements; and likewise the empty set is characterized by having no elements. John von Neumann reportedly said, "You don't understand math. You just get used to it." The empty set is just one of those things. You can't use your common sense to wrestle with it, that way lies frustration.
Quoting Metaphysician Undercover
You keep equivocating specification by predicates, on the one hand; and the axiom of extensionality, on the other. Every set is entirely characterized by its elements. Secondly, quite separately from that fact, are various ways of showing the existence of sets. Given a collection of sets we can take their union; or their intersection. Given a set we can take its powerset. Given a set and a predicate we can use the axiom of specification to obtain a subset of the original set consisting of exactly those of its members satisfying the predicate. There's also the axiom of replacement, and the axiom of choice. So first, a set is entirely characterized by its elements. And secondly, we have a toolbox for showing the existence of various sets: union, intersection, powerset, specification, replacement, choice. It's sort of like knowing what a house is, then learning how to build one. There's no ambiguity. Those are two separate things.
Quoting Metaphysician Undercover
Of course all the members of the Vitali set are real numbers. As are all the elements that are NOT members of the Vitali set. So you can have your point, but it's rather pointless. It doesn't do you any good.
Quoting Metaphysician Undercover
Well that's fine, then you believe in nonconstructive sets. You are willing to take the Vitali set and its members at face value. That's great. But you can see that it's very different than the set of prime numbers. With the set of prime numbers, we can look at a given individual number and say, "Yes you're in the club," or "No you're not in the club." With the Vitali set, there is no way to do that.
Quoting Metaphysician Undercover
They're not logically inconsistent, they're different ways of building sets; just as using brick or using wood are two different ways of building houses.
Quoting Metaphysician Undercover
Yes. We can do one or the other. We can talk about the set of prime numbers, in which we can talk about the entire set AND determine exactly which natural numbers are allowed into the set; and we can talk about the Vitali set, where we can NOT determine for any particular real number whether it belongs in the set or not.
Two different ways of obtaining or showing the existence of sets. There are lots of different ways of building houses and lots of different ways of building sets. I don't know why you think this is a problem.
Quoting Metaphysician Undercover
No, I only see various ways of building sets. Unions, intersections, powersets, specification, replacement, and choice. I believe those are all the set-building or set-existence tools. Like styles of houses or perhaps construction techniques or different choices of materials.
Quoting Metaphysician Undercover
The axiom of extensionality provides for the empty set. It's the set with no elements.
Quoting Metaphysician Undercover
Aha! You're on the verge of getting it. The set of pink flying elephants is an empty set. The set of people on the moon is an empty set. And the axiom of extensionality says that these must be exactly the same set. Because an object is an element of one if and only if it's an element of the other. I hope you'll take a moment to work through the logic. There is only one empty set, because the axiom of extensionality says that if for every object, it's a person on the moon if and only if it's a pink flying elephant, that the two sets must be the same.
Quoting Metaphysician Undercover
We know from the law of identity that everything is equal to itself. So what is the set of all things that are not equal to themselves? It's the empty set. And by the axiom of extensionality, it's exactly the same as the set of pink flying elephants and the people on the moon.
We notate this as [math]\{x : x \neq x\}[/math]. It's read: "The set of all x such that x is not equal to x." As an expert on the law of identity you will agree that there are no such x that satisfy that condition. So we have specified the empty set.
Quoting Metaphysician Undercover
I hope my more detailed explanation was better.
Quoting Metaphysician Undercover
Ok.
Quoting Metaphysician Undercover
You're confusing two different things. The axiom of extensionality tells you when two sets are identical: Namely, when they have exactly the same elements.
The axiom of specification, the powerset axiom, the axiom of choice, the axiom schema of replacement, and the axioms of union, intersection, and pairing (I forgot to mention that one) are all axioms that tell us which particular sets exist.
So we have an axiom that tells us when two sets are equal. That's extensionality, or "a set is entirely characterized by its elements." And we have a toolbox of ways to show that various sets exist. The're not in conflict with each other, any more than saying what a house is, is in conflict with the various construction techniques and styles of houses.
Quoting Metaphysician Undercover
No. The axiom of extensionality tells us when two sets are the same; namely, when they have exactly the same elements. The other axioms are a toolbox for knowing which sets exist. Specification is one of the tools as are union, intersection, etc.
Quoting Metaphysician Undercover
The empty set satisfies the symbolic expression I discussed earlier, whether or not it satisfies the English-language version. I can only refer you to the Wiki page on the axiom of extension, which I keep pointing you to and you keep not reading.
Quoting Metaphysician Undercover
Most definitely. The empty set is a set. No set is no set.
Quoting Metaphysician Undercover
Yes. You are confusing the axiom of extension, which tells us when two sets are the same, with the other axioms that give us various ways to build sets or prove that various sets exist.
I'm talking about stream objects in computer science, or equivalently the isomorphism
S <----> 1 x S in the Set category, where S is a stream object, 1 is the terminal object, i.e the singular set, and x is the Cartesian product. Unfolding the definition:
S <----> 1 x S <----> 1 x (1 x S) <----> 1 x (1 x (1 x S)) ....
Since each of the arrows is invertible, S clearly has, by definition, a surjection onto any finite set, which is what i meant by Kuratowski infiniteness.
On the other hand, recall that in category theory every element of a set S is an arrow of the form
1 --> S. However, these arrows haven't been specified in my above definition of S, and therefore the number of elements of S is currently zero, i.e. S is the empty set, which is another way of saying that S is a completely undefined set until the first observation is made.
Every time an observation is made, an arrow of the form 1--> S is introduced into the above category, and we can denote the current state of the stream by shifting from left to right in the above diagram. But at every moment of time, the arrow S --> S that is implicit in the the product S ---> 1 x S is a bijection, meaning that is S is forever dedekind-finite.
S cannot exist as an internal set of ZF because it isn't a well-founded set, although it can exist in the sense of an "external set" that is to say, inside ZF as part of a non-standard interpretation of an internal well-founded set. And it it cannot exist in any capacity inside ZFC.
All of which is tantamount to saying that ZF has only partial relevance to modern mathematics in terms of being an axiomatization of well-foundedness, whilst ZFC is completely and utterly useless, failing to axiomatize the most rudimentary notions of finite sets as used in the modern world.
You do not seem to be grasping the problem. If a set is characterized by its elements, there is no such thing as an empty set. No elements, no set. Do you understand this? That is the logical conclusion we can draw from " a set is characterized by its elements". If we have no elements, we have no set. If you do not agree with this, explain to me how there could be a set which is characterized by its elements, and it has no elements. It has no character? Isn't that the same as saying it isn't a set?
So we cannot proceed to even talk about an empty set because that's incoherent, unless we dismiss this idea that a set is characterized by its elements. Can we get rid of that idea? Then we could proceed to investigate your interpretation of the axiom of extensionality, which allows you to say "If you have two sets such that they have no elements, they're the same set; namely the empty set", because "empty set" would be a coherent concept. Until we get rid of that premise though, that a set is characterized by its elements there is no such thing as a set with no elements, because such a set would have no identity whatsoever, and we could not even call it a set.
Quoting fishfry
You are not grasping the distinction between 'characterized by its elements', and 'characterized by its specification' which I'm trying to get though to you. When you say "the set of all purple flying elephants", this is a specification, and this set is characterized by that specification. There are no elements being named, or described, and referred to as comprising that set, there is only a specification which characterizes the set.
Quoting fishfry
Where do you get this idea from? Clearly your example "the set of all purple flying elephants" is not characterized by its elements. You have made no effort to take elements, and compose a set You have not even found any of those purple flying elephants. In composing your set, you have simply specified "purple flying elephants". Your example set is characterized by a specification, not by any elements. If you do not want to call this "specification", saving that term for some special use, that's fine, but it's clearly false to say that such a set is characterized by its elements.
This is what happens when we proceed deep into the workings of the imagination. We can take a symbol, a name like "purple flying elephants", or any absurdity, or logical incoherency, like "square circles", each of which we assume has no corresponding objects However, we can then claim something imaginary, a corresponding imaginary object, and we can proceed under the assumption that the name actually names something, a purple flying elephant in the imagination. You might then claim that this imaginary thing is an element which characterizes the set. But if you then say that the set is empty, you deny the reality of this imaginary thing, and you are right back at square one, a symbol with nothing corresponding. And so we cannot even call this a symbol any more, because it represents nothing.
Quoting fishfry
Now you've hit the problem directly head on. To be able to have an empty set, a set must be characterized by it's specification, as I've described, e.g. "pink flying elephants". So. the set of pink flying elephants is one set, characterized by the specification "pink flying elephants", and the set of people on the moon is another set, characterized by the specification "people on the moon". To say that they are exactly the same set, because they have the same number of elements, zero, is nor only inconsistent, but it's also a ridiculous axiom. Would you say that two distinct sets, with two elements, are the exact same set just because they have the same number of elements? I think you'll agree with me that this is nonsense.
And to say that each of them has the very same elements because they don't have any, is clearly a falsity because "pink flying elephants" is a completely different type of element from "people on the moon". If at some point there is people on the moon, then the set is no longer empty. But the two sets have not changed, they are still the set of pink flying elephants, and the set of people on the moon, as specified, only membership has changed. Since the sets themselves have not changed only the elements have, then clearly they were never the same set in the first place.
Of course, you'll claim that a set is characterized by its elements, so it was never "the set of pink flying elephants in the first place, it was the empty set. But this is clearly an inconsistency because "pink flying elephants was specified first, then determined as empty. So that is not how you characterized these sets. You characterized them as "the set of pink flying elephants", and "the set of people on the moon".
If you had specified "the empty set", then obviously the empty set is the same set as the empty set, but "pink flying elephants", and "people on the moon" are clearly not both the same set, just because they both happen to have zero elements. The emptiness of these two sets is contingent, whereas the emptiness of "the empty set" is necessary, so there is a clear logical difference between them.
Quoting fishfry
I don't know why you can't see this as a ridiculous axiom. You say that a "person on the moon" is a "pink flying elephant". That's ridiculous.
Quoting fishfry
See the consequences of that ridiculous axiom? Now you are saying that a pink flying elephant is a thing which is not equal to a pink flying elephant, and a person on the moon is not equal to a person on the moon. Face the facts, the axiom is nonsensical.
Quoting fishfry
Obviously, the axiom of extension is very bad because it fails to distinguish between necessity and contingency.
This has murky and unclear relevance to what went before in the same post. I can't tell if you are trying to explain something to me or promoting an agenda. More clarity and less stridence would be helpful to me; but I'm not sure if being helpful is your intention.
What I mean is that one can look up the entry for the axiom of choice on nLab, without encountering a rant against ZFC. So I think you're the one adding that part, and not your fellow constructivists / category theorists / programmers or whatever direction you're coming from.
Indeed, nLab expresses choice as "every surjection splits," which they note means "every surjection has a right inverse," in set theory. This formulation is easily shown to be equivalent to the traditional statement of the axiom of choice. There is no distance between the category-theoretic and set-theoretic views of choice.
Rather than try to understand set theory on its own terms, you just want to fight with it. Why? I'm taking the trouble to explain it to you, on its own terms. I'm on record multiple times saying that I make no claims that it's "true" in any meaningful sense; or even meaningful in any meaningful sense. So why try to argue with me about the subject? The person you want to address your complaints to is [url=https://en.wikipedia.org/wiki/Ernst_ZermeloErnst Zermelo[/url], who more than anyone is responsible for the modern incarnation of the standard axioms of set theory. Cantor gets all the credit and Zermelo did the heavy lifting. Zermelo died in 1953 so he's not available for you to complain to; but I am not available for you to complain to either. I"m not defending the truth, meaning, sanity, or sense of set theory. I'm only describing to you how it is. You will have to take your complaints elsewhere.
Perhaps this is a fundamental confusion on your part, but I don't see why. I have explained my viewpoint many times. I'm describing set theory to you. I am not defending it, not advocating it, not promoting it. But I am, to the best of my ability, giving an accurate account of the basics of the subject as it is understood by mathematicians. So if you want to learn it, let's proceed. If you just want to argue with me about it, that makes for a one-sided and tedious interaction.
Let me say this once and for all: I am not the lord-high defender of set theory. It's exactly like chess. I'm teaching you the rules. If you don't like the game, my response is for you to take up some other game more to your liking.
Quoting Metaphysician Undercover
You're too hung up on the empty set. As I said in my previous post, students generally have a hard time getting accustomed to vacuous arguments. It's perfectly analogous to the subject of material implication in logic. Students don't get why "If 2 + 2 = 5 then I am the Pope" is a true statement. At some point most of them get it, and some never do.
Quoting Metaphysician Undercover
No. It's not only that you're wrong, but it's a nothingburger of an issue. It's like a beginning logic student constantly arguing with the professor about material implication. The facts of the matter are never going to change. The student can only accept it, or drop the course and enroll in a different one more to his or her liking. One tactic is to just accept it ("it" being whatever is bothering the learner at the moment) on faith, keep working at the subject, and one they they'll wake up and realize that it's all perfectly obvious and they can't even remember a time when it wasn't. That's the tactic I recommend to you.
Quoting Metaphysician Undercover
But I have already explained to you in my previous post, that "a set is characterized by its elements" is merely an English-language approximation to the axiom of extentionality, which actually says, [math]\forall A \forall B (\forall X (X \in A \iff X \in B) \implies A = B)[/math]. That is the axiom that says that two sets are equal if they have exactly the same elements. And by a vacuous argument -- the same kind of argument that students have had trouble with since logic began -- two sets are the same if they each have no elements.
If you won't grapple with the symbology, you have to accept it on faith. You can't just fall back on the imprecise English-language version, now that I've shown you (twice) the formal version.
Quoting Metaphysician Undercover
You're being tedious. First, the matter is trivial. The empty set is a thing in set theory. You can't allow your learning to be stuck on this one point. Accept it and move on, or go find something else to be interested in. Secondly, if you will put in the work to understand the symbology as described on the Wiki page for the axiom of extensionality, at some point you'll probably just get it.
Quoting Metaphysician Undercover
The formal symbolic expression of extensionality could not be more clear. The fact that you won't engage with it is not my problem.
Quoting Metaphysician Undercover
Perhaps. But again, you're trying to learn a subject, and every time you're shown one of the basic principles, you just want to argue. You make it difficult on yourself. Once you learn basic set theory, you can set about developing an alternative version if you like. Einstein changed physics, but before he did that he mastered classical physics. Right? Right.
Quoting Metaphysician Undercover
You're just being tedious. The formal symbology is perfectly clear. And even if it isn't clear to you, you should just accept the point and move on, so that we can discuss more interesting things. You're causing yourself to get stuck on a relatively minor point. You have two ways out: One, grapple with the formal symbology here. Two, accept it and move on. Repeating the same tired and fallacious objections is no longer an option, at least with me. Maybe you can get someone else to play.
Quoting Metaphysician Undercover
Here you have made up your own phrase, "characterized by its specification." I have not said that. The axiom of specification is one of the axioms that tell us when particular sets may be said to exist. I explained this to you in painful detail in my previous post, and you are just ignoring what I said. The axiom of extensionality tells us when two sets are the same. The axiom schema of specification tells us when certain sets exist. There are other set existence axioms: pairing, union, intersection, powerset, replacement, and choice. Frankly if you want to complain about axioms, it's replacement you should be concerned about. It's very murky. Zermelo's original formulation didn't even include it, as I've recently learned due to @TonesInDeepFreeze's repeated mention of Zermelo set theory, or Z. I went and looked it up and learned something new.
Quoting Metaphysician Undercover
If by specification you mean predicate, then "E(x) = x is a flying elephant" is most definitely a specification. And in the axiom of specification, that's what's meant. The fact that a predicate may have an empty extension is not a bug, it's a feature.
Quoting Metaphysician Undercover
From the axiom of extensionality. You know, I don't care if you believe in the empty set or not. But after my having directedyour attention to the axiom of extensionality so many times, I don't see how you can ask where I got the idea. It's one of the axioms.
Quoting Metaphysician Undercover
Accept it and let's move on; or don't accept it and quietly seethe while we move on. I can't engage with you on this point anymore. I've already explained it. The axiom of specification allows us to use a predicate to form a set. The predicate is not required to have a nonempty extension.
Quoting Metaphysician Undercover
As I'm always fond of pointing out, the unit circle in the taxicab metric is a square. There's a picture of a square circle at this link. How about married bachelor, that's a better example.
Quoting Metaphysician Undercover
It's merely a predicate with an empty extension.
Quoting Metaphysician Undercover
What's true is that given any thing whatsoever, that thing is a pink flying elephant if and only if it's a person on the moon. So the axiom of extensionality is satisfied and the two sets are equal. If you challenged yourself to work through the symbology of the axiom of extentionality this would be perfectly clear to you.
Quoting Metaphysician Undercover
Of course not. It's not a matter of cardinal equivalence. The elements themselves have to be respectively equal. {1,2} and {1,2} are the same set. {1,2} and {3,47} are not.
Quoting Metaphysician Undercover
It would be false. It would not be nonsense, that's a value judgment. And your value judgments regarding mathematics are not good.
Quoting Metaphysician Undercover
You are right about that. But that's because we are making up examples from real life. Math doesn't have time or contingency in it. 5 is an element of the set of prime numbers today, tomorrow, and forever. The "people on the moon" example was yours, not mine. I could have and in retrospect should have objected to it at the time, because of course it is a temporally contingent proposition. I let it pass. So let me note for the record that there are no temporally contingent propositions in math.
Quoting Metaphysician Undercover
It's a bad example because one of your propositions is temporally contingent. I noted that at the time you mentioned it but let it pass. I see that was a mistake. I have now rectified my error. There are no temporally contingent propositions in math.
Quoting Metaphysician Undercover
The two sets, assuming that we mean at the present moment, are the same, namely the empty set, because the condition in the axiom of extensionality is satisfied. A thing is in one of those sets if and only if it's in the other. Therefore the sets are the same. That's all there is to it.
Quoting Metaphysician Undercover
Only by virtue of the people on the moon being temporally contingent. So it's a bad example, which I should have pointed out when you first mentioned it.
Quoting Metaphysician Undercover
Nobody says that. What is true is that the axiom of extensionality is satisfied. Until you roll up your sleeves and put in the work to understand that, you'll spin yourself in circles.
Quoting Metaphysician Undercover
You are being tedious. You wrote an entire post on the nonsense. Go understand what the axiom of extensionality says. You're running yourself in circles because you can't be bothered to challenge yourself to work through what the axiom says.
Quoting Metaphysician Undercover
Nobody is saying a pink flying elephant is a thing. You're just a logic student having trouble with vacuous arguments. Put in the work to understand it, or just accept it and move on. Being endlessly tedious, writing an entire post about your own misconceptions, is pointless.
Quoting Metaphysician Undercover
There are no temporally contingent propositions in math. The people on the moon example is a bad one for that reason. I was wrong rhetorically to let it pass without objection earlier, because now you just want to use it to make a sophistic point.
Bottom line is that the empty set is a purely formal object that satisfies some formal conditions. It's not "real" and it's not helpful to try to understand it in terms of common sense. The following SE thread, in particular the checked answer, may be helpful.
https://philosophy.stackexchange.com/questions/14823/why-do-we-have-a-problem-about-understanding-the-concept-of-the-empty-set
Category theory is a useful meta-language for understanding precisely where the mathematical foundation proposed in the early 20th century goes wrong, as well as being helpful for relating alternative theories. CT is itself philosophically neutral in the sense that it only assumes the presence of identity arrows in a category, but places no other constraints on either the presence or absence of arrows, provided the laws of arrow composition and association are obeyed. Therefore disputes between intuitionists, formalists and platonists carry over into the language.
Like a mathematics department, nlab as an encyclopedia is obviously going to disseminate mathematics in a politically neutral fashion. Or perhaps i should have said, unlike a mathematics department. But political neutrality doesn't amount to reasonableness regarding which mathematics should be prioritised.
I'm a philosopher, my game is to analyze and criticize the rules of other games. This is a matter of interpretation. If you do not like that, then why are you participating in a philosophy forum?
As much as you, as a mathematician are trying to teach me some rules of mathematics, I as a philosopher am trying to teach you some rules of interpretation. So the argument goes both ways, you are not progressing very well in developing your capacity for interpreting. But if you do not like the game of interpretation, then just do something else
Quoting fishfry
This is why the axiom of extensionality is not a good axiom. It states something about the thing referred to by "set", which is inconsistent with the mathematician's use of "set", as you've demonstrated to me.
Quoting fishfry
We've already been through this problem, a multitude of times. That two things are equal does not mean that they are the same. That's why I concluded before, that it's not the axiom of extensionality which is so bad, but your interpretation of it is not very good. But I now see that the axiom of extensionality is itself bad.
Quoting fishfry
In case you haven't noticed, what I am interested in is the interpretation of symbols. And obviously the symbology of the axiom is not perfectly clear. If you can interpret "=" as either equal to, or the same as, then there is ambiguity.
Quoting fishfry
Actually, I'm starting to see that this, what you claim in your vacuous argument, is not a product of the axiom of extensionality, but a product of your faulty interpretation. By the axiom of extensionality, a person on the moon is equal to a pink flying elephant, and you interpret this as "the same as". So the axiom is bad, in the first place, for the reasons I explained in the last post, and you make it even worse, with a bad interpretation.
Quoting fishfry
You really do not seem to be getting it. If, we can "use a predicate to form a set" as the axiom of specification allows, then it is not true that a set is characterized by its elements. It's characterized by that predication. The two are mutually exclusive, inconsistent and incompatible. Specification allows for a nonempty set, I have no problem with this. But to say that this set is characterized by its elements is blatantly false. It has no elements, and it is characterized as having zero elements, an empty set. So it's not characterized by its elements, it's characterized by the number of elements which it has, none. .
Quoting fishfry
Yes, this is the problem with the axiom of extension, in its portrayal of the empty set. It is saying that if two specified sets each have zero elements, then "the elements themselves" are equal. However, there are no such elements to allow one to judge the equality of them. So there is no judgement that "the elements themselves" are equal, because there are no elements to judge, and so the judgement of cardinal equivalence, that they have the same number of elements, zero, is presented as a judgement of the elements themselves.
You ought to recognize, that to present a judgement of cardinal equivalence, as a judgement of the elements themselves, is an act of misrepresentation, which is an act of deception. I know that you have no concern for truth or falsity in mathematical axioms, but you really ought to have concern for the presence of deception in axioms.
Quoting fishfry
Now, do you agree, that when there are no elements, it makes no sense to say that the elements themselves are respectively equal? What is really being judged as equal is the cardinality. They both have zero elements.
Quoting fishfry
No, the axiom of extensionality does not tell us when two sets are the same, that's the faulty interpretation I've pointed out to you numerous times already, and you just cannot learn. It tells us when two sets are equal.
That faulty interpretation is what enables the deception. Equality always indicates a judgement of predication, and in mathematics it's a judgement of equal quantity, which you call cardinal equivalence. When you replace the determination of the cardinality of two empty sets, "equal", with "the same", you transfer a predication of the set, its cardinality, to make a predication of its elements, "the same as each other". I believe that's known as a fallacy of division.
Quoting fishfry
Well, "pink flying elephants" was your example, and it's equally contingent. The issue of temporally contingent propositions raises a completely different problem. The only truly necessary empty set is the one specified as "the empty set". As your examples of square circles and married bachelors show, definitions and conceptual structures change over time, so your assertion that mathematics has no temporally contingent propositions is completely untrue. It may be the case that "the empty set" will always refer to the empty set, necessarily, but how we interpret "empty" and "set" is temporally contingent. So temporal contingency cannot be removed from mathematics as you claim. This is the problem of Platonic realism, the idea that mathematics consists of eternal, unchanging truths, when in reality the relations between symbols and meaning evolves.
You reject general relativity and set theory because of your ontology, and your ontology rejects emergence. There is the problem. An object has no weight on it's own, and neither does spacetime. But together they form the world of substance we experience as weight. In set theory Zero means "no thing" and Set is a collecting of something. But the concepts together form something that is useful in the practice of set theory
The "further reading" is good too
The axiom is:
Axy(Az(zex <-> zey) -> x = y)
that is the sufficiency clause.
The necessity clause comes from identity theory:
Axy(x = y -> Az(zex <-> zey))
So together we have the theorem of sufficiency and necessity:
Axy(Az(zex <-> zey) <-> x = y)
/
'set' is not a primitive of set theory. However it, may be defined in the language of set theory:
df: x = 0 <-> Ay ~yex
df: x is a class <-> (Ey yex v x=0))
df: x is set <-> (x is a class & Ey xey))
That definition also carries over to class theory such as Bernays class theory (more commonly known as 'NBG set theory').
But formally 'is a set' does not occur, rather it is an English phrase to stand for a formal predicate symbol. While 'set' is used to suggest our pre-formal notion of sets, it is not formally required. We could use 'zet' 'Zset' or myriad other words. Same for the phrase 'empty set'. So, while one may choose to argue that the notion of an empty set does not adhere to our usual understanding of what sets are, one should be careful not to charge that set theory is inconsistent on that basis.
The theorem and definition at issue are:
th: E!x Ay ~yex
df: Ax x = 0 <-> Ay ~yex.
No mention of 'set' or 'empty' or 'empty set'.
Which math gets prioritized is of course a matter of historical contingency. Set theory in the 20th century, maybe category theory / type theory / topos theory / whatever in the 21st, and who knows what in the 22nd. I agree with you about this. However, you have expressed an antipathy to the axiom of choice that is not shared by category theorists. I'm not exactly sure where you are coming from. Are you a constructivist? They are gaining mindshare these days through the influence of computers. But Turing showed us that there are easily-stated problems that can not possibly be solved by a computer. So there will always be a place for nonconstructive math. In my "inappropriately trained" opinion, of course.
Quoting sime
I got a chuckle out of this.
Campus or departmental politics regarding subject matter and its relative "importance" can depend upon current interests of the faculty - and grant money. Sometimes a group of faculty convince administration to focus on or emphasize a particular topic those mathematicians are eager to pursue. Since math is a social activity directions may be decided by social interaction - and strong personalities. And, not least among the reasons for these choices, a department may decide to build around a well-known and accomplished colleague. The awarding of grant money is another story, similar to the above.
Category theory seems topical these days. My old clique - largely kerpunkt - hoped for a resurgence of interest in analytic continued fraction theory. The leaders have simply passed on. RIP
A philosophy crank is more like it. You have zero familiarity with the 20th century literature on the philosophy of set theory. You haven't read Maddy, Quine, or Putnam. You have no interest in learning anything about the philosophy of set theory. When I mentioned to you the other day that Skolem was skeptical of set theory as a foundation for math, you expressed no curiosity and just ignored the remark. Why didn't you ask what his grounds were? After all he was one of the major set theorists of the early 20th century.
You don't seem to be able to engage in logical reasoning. You keep saying things that are demonstrably false, and when corrected, you simply repeat the same mistakes in your next post. You don't seem to understand elementary sentential logic, for example material implication and logical equivalence. About this more in a few paragraphs.
Quoting Metaphysician Undercover
What do you think of the knight move in chess? When pressed, you tried to claim it's "physical" because chess sets are made of atoms. Childish sophistry.
Quoting Metaphysician Undercover
I'm here discussing the philosophy of math. I hope you don't think anything we've discussed is actual math. Do you think mathematicians sit around and talk about whether the empty set exists? And as I say, I know a little bit about the actual philosophical literature on the subject, and you haven't the slightest interest in it.
Quoting Metaphysician Undercover
Is that why you can't work your way through a simple logical equivalence in sentential logic?
Quoting Metaphysician Undercover
Your cranky and ignorant ideas of math aren't subject to interpretation, only derision.
Quoting Metaphysician Undercover
Oh I'll be here to discuss the philosophy of math, if anyone is interested. Sadly you have nothing intelligible to say on the subject.
Quoting Metaphysician Undercover
Nonsense. It's your own inability to follow an elementary exercise in logic that keeps you stuck.
Quoting Metaphysician Undercover
That's not relevant here. However, two things that are mathematically equal are indeed the same.
Quoting Metaphysician Undercover
Mine is the perfectly standard interpretation, comprehensible to everyone who spends a little effort to understand it. Two sets are the same if and only if they have the same elements. Formally, if a thing is in one set if and only if it's in the other; which (as we will shortly see) includes the case where both sets are empty.
Quoting Metaphysician Undercover
Like I say, you need to take that up with Ernst Zermelo. Or the authors of every set theory text in the world.
Quoting Metaphysician Undercover
How can that be? You are completely unable to understand even the most elementary symbolic reasoning.
Quoting Metaphysician Undercover
Are you saying that two things can be "the same" but not equal? Are you sure whatever you're on is legal in your jurisdiction?
Quoting Metaphysician Undercover
I'm going to walk you through this.
First, do you understand material implication? Material implication has the following truth table:
In other words "If 2 + 2 = 5 then I am the Pope" is a true material implication. Do you understand that? Do you agree? Do you have a disagreement perhaps?
Ok. Let [math]A[/math] be the set of pink flying elephants, and [math]B[/math] the set of people on the moon. For purposes of this exercise, let's assume these are not contingent. If you can't do that then make them the set of even numbers not divisible by 2, and the set of primes with nontrivial factorizations if you like.
Now I claim that for all [math]X[/math], it is the case that [math]X \in A \iff X \in B[/math]. That is read as, "X is an element of A if and only if X is an element of B.
In sentential logic we break this down into two propositions: (1) If X is an element of A then X is an element of B; and (2) If X is an element of B then X is an element of A."
Now for (1). If X is a pink flying elephant, then it's a person on the moon. Is that true? Well yes. There are no pink flying elephants and there are no people on the moon. So this is line 4 of the truth table, the F/F case, which evaluates to True. So (1) is true.
How about (2)? Well the argument is exactly the same. If X is a person on the moon, then they are a pink flying elephant. As in (1), this is the False/False case of material implication.
Having shown both directions of the implication, we have that A and B are logically equivalent.
The axiom of extensionality says that if it's the case that X is in A if and only if X is in B, then A = B.
Therefore, by an exercise in elementary sentential logic, we see that the axiom of extensionality says that the two sets are the same.
Now here is what I know. I know that you are totally incapable of following this simple chain of logic. Or perhaps just unwilling. Either way, it's no longer my problem.
Quoting Metaphysician Undercover
Of course it is. We can use specification to prove the existence of some set; AND by extension if that set has the same elements as some other set, then the two sets are the same.
Quoting Metaphysician Undercover
No no no. "Characterized" in this context is YOUR word but it's not what I've said and not what set theory says. Set theory says that (1) we can show some set exists using specification; and that set is also subject to extensionality: if some other set has the same elements, then the two sets are the same.
For example the set of "the first three positive integers is {1,2,3}. The set of the positive square roots of 1, 4, and 9 is (1, 2, 3}. Two distinct specifications. But the two sets have exactly the same elements; so they are the same set.
Specification lets you show some set exists. Extensionality tells you when some set is equal to some other set.
I can not for the life of me figure out why you won't get this. I will say that I'm not sure how we got onto this particular subtopic, and that I'm finding it tedious in the extreme, so I probably won't be replying back much as long as you continue to be trollishly repeating this fallacious line of argument.
Quoting Metaphysician Undercover
No, they are quite independent of each other. Specification is one way (out of several others) to show that a given set exists. Extensionality tells you when that set is equal to some other set.
Quoting Metaphysician Undercover
I just gave you a formal proof to the contrary. An object is in A if and only if it's in B. If A and B happen to be empty, that is a true statement. Therefore by extensionality, A = B. Or in words, "There's only one empty set."
Quoting Metaphysician Undercover
You're confusing yourself because you are unable/unwilling to do a little basic sentential logic, or perhaps that you do not understand material implication.
Which is it?
Quoting Metaphysician Undercover
You are confusing yourself because you can't/won't follow the symbolic logic. For all X, X is in A if and only if x is in B. That's as clear as can be.
Tell me, do you disagree or happen to not know basic sentential logic? Do you understand that "if 2 + 2 = 5 then I am the Pope" is true?
Quoting Metaphysician Undercover
No middle e in judgment. Please make a note of it.
Quoting Metaphysician Undercover
X is in A if and only if X is in B. Can it be the case that you don't actually know basic sentential logic, and don't understand material implication, and therefore cannot understand the content of the axiom of extensionality? That would explain a lot.
Quoting Metaphysician Undercover
You interpret your own ignorance as deception by others. Pretty funny.
Quoting Metaphysician Undercover
They are vacuously respectively equal. I see that it must be the case that you do not actually understand material implication. That explains quite a lot.
The axiom of extensionality, X is in A if and only if X is in B, is the same as "If 2 + 2 = 5 then I am the Pope; AND if I am the Pope then 2 + 2 = 5." I see that you truly don't get this. @Meta my friend you need to go study up on basic sentential logic. As a self-proclaimed philosopher you are missing the very basics.
Quoting Metaphysician Undercover
I see why you think this. It's because you don't understand material implication and logical equivalence.
Quoting Metaphysician Undercover
Someone a while back pointed out that in math, equality is extensional. You are taking it intentionally. But whatever. It's not relevant to the fact that we've discovered that you can't do basic sentential logic. That's something you should remedy at once.
Quoting Metaphysician Undercover
Throwing more crap on the wall doesn't remedy your inability to understand a simple logical argument.
Quoting Metaphysician Undercover
For purposes of this discussion, we take the two predicates as absolute and not contingent. You're just raising this red herring to sow confusion. The only one confused here is you.
Quoting Metaphysician Undercover
You can throw all the crap on the wall you like. I'm done cleaning it up. Change the subject, I'm done with this. Go read a book on logic and then work through the axiom of extensionality, which frankly is quite simple.
If some of this is relevant to the points I've made, then provide some quotes or references. Otherwise what's the point in mentioning something which is not relevant?
Quoting fishfry
You haven't addressed the point. To have "the same elements" requires a judgement of elements. Having no elements is not an instance of having elements, and there are no elements to be judged. An empty set has no elements therefore two empty sets do not have the same elements, because they both have no elements. Therefore two empty sets are not the same.
Quoting fishfry
We've been through the law of identity before, and you still show no desire to understand it.. No two things are the same, according to that law. If it's "the same", then there is only one thing. That's what "the same" refers to according to the law of identity, one and the same thing. The law of identity dictates that we use "same" to refer to only one thing, so it is impossible that two distinct things are the same. However, two distinct things may be equal. Therefore "equal" is not synonymous with "the same".
Quoting fishfry
Sorry fishfry, but you'll need to do a better job explaining than this. Your truth table does not show me how you draw this conclusion.
Quoting fishfry
You have already determined that there are no elements in both sets A and B. This is predetermined, they are empty sets. So your starting point, "(1) If X is an element of A then X is an element of B" is not relevant, there are not elements. That's like saying if C and D are both green, when you've already determined that they are not green. It's an irrelevant premise, and your entire appeal to material implication is unacceptable. We already know that there are no elements of both A and B, so that premise concerning the elements of A is not applicable. The two premises "A is an empty set", and "if X is an element of A" are fundamentally contradictory.
I'll give you credit for at least addressing the point now. It was a nice try, but your attempt is a failure.
Quoting fishfry
As explained above, it's actually an irrelevant, and inapplicable statement. And it can only be applied under contradiction. When a set has been determined as empty, then to talk about objects within that set is contradiction. So an attempt to apply this statement to empty sets is contradictory. Look at what you're saying 1)There are not any objects in set A. 2) An object is in A if... See the contradiction? When you've already designated A as having no objects, how does it make sense to you to start talking about the condition under which there is an object in A? Do you agree that there is contradiction here?
As you can see, your attempt at a formal proof is a failure due to contradicting premises.
Quoting fishfry
The thing with this type of deception, is that you can either recognize it as deception, and reject it, or you can join it, and become one of the deceivers. This is why mathematics is similar to religion (Tones will disagree), the authors have good intentions, but once falsity is allowed into the premises, deception is required to maintain respect for the premises amongst the masses. When the deception has been pointed out to you, as I have, then you can either reject it and work toward dismantling the system which propagates it, or you can support it with further deception. You it appears, are choosing to be one of the deceivers.
Quoting fishfry
This thoroughly supports my argument. If the empty sets are necessarily empty, absolute and not contingent, then to talk about the conditions under which there are elements in those sets is very clearly contradiction.
Like for example, when someone says "an empty set has no elements", and also says "the elements of the empty set A are the same elements as the elements of the empty set B". The former says the empty set has no elements, the latter states its negation "the elements of the empty set...".
Or, even if one were to say that an empty set has no elements, absolute and not contingent, and then states the conditional "if X is an element of the empty set...", as if the set is only contingently empty, that would also constitute a statement and its negation.
Don't you agree?
P & ~P
where P is any statement.
A conjunction of a statement and its negation in the language of set theory is of the form
P & ~P
where P is a formula in the language of set theory.
Set theory is inconsistent if and only if there is a such a conjunction that is a theorem of set theory.
I'm going to let you have the last word. I'm out. But for the record, can you please name the specific individuals involved in this deception? We need their names to hold them accountable at the Stalinist show trials to begin soon. Cantor? Zermelo? Mrs. Zermelo, who was pro choice? Abraham Fraenkel? Should John von Neumann be included? He did invent mathematical economics and worked on the hydrogen bomb, but ... he DID do foundational work in set theory as well. Might was well include him on the list. How about the modern set theorists Solovay, Magidor, Shelah? Or the contemporary ones like Woodin and Hamkins? The modern philosophers of set theory like Quine, Putnam, and especially Maddy? Are they all involved in this deception? Please be specific, we need to know how many cells to reserve at Gitmo.
Quoting Metaphysician Undercover
You don't know material implication? You will find "my" truth table on that page. How deep exactly is your ignorance? "My" truth table? Do you really mean to say you never saw this before? I guess I don't understand how that could be. Honestly, in all your time on this forum and presumably studying the philosophy of math, you never saw basic sentential (aka propositional) logic?
Here you go. https://en.wikipedia.org/wiki/Propositional_calculus
You can have the last word. Though after calling modern mathematics a "deception" and admitting that you are unfamiliar with the truth table for material implication, I don't see how you could top what's gone before.
Thanks for the chat. All the best.