There are a few different conceptions of infinity in mathematics.
There's what the usual infinity symbol represents: [math]\infty[/math], which usually denotes a limiting process: [math]\lim_{x\rightarrow\infty}f(x)[/math], IE what value [math]f(x)[/math] tends to when [math]x[/math] becomes arbitrarily large. Formally this corresponds to a definition of a limit and can be considered shorthand for it.
Then you've got cardinal numbers, which count how many of something there are. The smallest infinite cardinal is called [math]\aleph_0[/math], which is the size of the set of natural numbers [math]\mathbb{N}=\{0,1,2,3,...\}[/math]. Then there are ordinal numbers, which agree with cardinal numbers up to [math]\aleph_0[/math] and can disagree beyond that - they correspond to different ways of ordering infinite sets of things. For example, the standard ordering of [math]\mathbb{N}=\{0,1,2,3,...\}[/math] is given the symbol [math]\omega[/math], which denotes its order type. If you removed 42 from [math]\mathbb{N}[/math] and stuck it on the end (after the infinity of numbers), you'd have the same set of elements but it would look like [math]\{0,1,2,...,41,43,44,...,42\}[/math], and this is given the order type [math]\omega+1[/math]. You can separate out the odds and evens similarly and end up with [math]\omega+\omega=2\omega[/math]. This operations allow you to define standard arithmetic operations on infinities relating to orders, and similarly for cardinals.
In the first case, infinity is a shorthand for a limiting process (the infinity is hidden in the quantifier 'for all epsilon'), in the second case infinite objects are referred to explicitly.
In the first case it's easier to think of as a direction. In the second case - for cardinals - they give the size of infinite sets, so yes they are probably quantities since they represent the magnitude of something.
Gustavo FontinelliMarch 11, 2018 at 17:06#1611120 likes
Following the "default definition", quantity stands for the magnitude of countable and reducible things. I mean, in a geometric view, would be like distance, the space between the initial and the final point. When you're counting something, you're presuming that there's a limit and when you reach the limit you'll be known the quantity.
Also, a x quantity of filler stuff fill in a x quantity of fillable things.
For example, two shoes fits in a box, if you increase the quantity of shoes to 3 shoes, it won't fit any more because it crosses the limit, it can also happen in the negative way. The thing is that with you have an infinitely large box, with an infinite amount of shoes in, no matter how many shoes you take off from the box, it won't change nothing. So talking about quantity doesn't make sense any more.
"Limiting processes" tend to have a somewhat uneasy relationship with the axioms of Set Theory and Peano Arithmetic which underlie damn near everything about number theory. If you are talking about aleph-null infinities then, of course, every aleph-null infinity has a precise numeric value (though this value is impossible to identify).
Not at all; I am trying to tread a delicate line between "mathematics" and "mathematical philosophy". Most non-mathematicians, and even many mathematicians, conceive of mathematics as the quintessential, monolithic embodiment of perfect rationalism, the ultimate logical system; and of course, it isn't. There are many logical grey areas, even at basic levels. For example, is 0.9... equal to 1? Or is it the largest real number which is less than 1? There are persuasive mathematical arguments on both sides.
"The wise man doubts often, and his views are changeable; the fool is constant in his opinions, and doubts nothing, because he knows everything, except his own ignorance" (Pharaoh Akhenaton).
Ok. Well epsilon-N implies 0.9 recurring = 1 anyway. AFAIK it's even true in non-standard analysis. 1 - infinitesimal isn't the same thing as 0.9 recurring.
No, infinity is not a quantity it is a direction on any scale in which it is listed as a measurement.
East is not a location, destination, or even an obtainable goal. It is a direction relative to the current position and might more properly be stated as "east of here," wherever here may be, in the same sense that x + 1 is not an absolute quantity but instead something greater than x.
Count Radetzky von RadetzMarch 14, 2018 at 12:09#1618320 likes
From a Hegelian perspective, I would rationally perceive infinity not as a quantity.
1. A quantity is a specified amount of something. It has a limit. The infinite is that which has no limits and so cannot be quantified. Therefore, not a quantity as not quantifiable.
This is just... no. Look, even if I take your definition of quantity, I can easily show infinity is a quantity. Take the set of Natural Numbers (o, 1, 2, 3...). In set theory, the concept of "size" is formalized as what is known as "cardinality". The cardinality (size) of the set of Natural Numbers is infinity, specifically aleph-null. QED. You can say the Natural Numbers have "no limit" in the sense that it can always get bigger, but that doesn't mean it's impossible to quantify.
2. Infinity is not limited to numbers (because it has no limit). if you say infinity is only a number you have broken the law of none contradiction as you have put a limit on something defined as having no limits. Therefore, infinity contains numbers but numbers do not contain infinity as numbers are limited to number.
A better way to think about it is there are different kinds of infinite numbers, some larger or smaller than others. The set of Real numbers, for instance, is a larger infinity than the infinity of the Natural numbers. Cantor proved this with a proof by contradiction. No one is contradicting themselves saying there are infinite quantities.
GreenPhilosophyJune 18, 2018 at 03:03#1889140 likes
You can add 1 to any real number, so infinity isn't a real number. Infinity is a concept.
Colloquially, infinite is a quantity that's not a number, [math]|\mathbb{R}| \sim \infty \notin \mathbb{R}[/math].
But it's ambiguous (hence the [math]\sim[/math]).
As it turns out there's more than one infinite, there are infinitely many different infinites, no less (Cantor).
Anyway, Dedekind and Tarski came up with different (general) definitions that can be shown equivalent.
You can separate out the odds and evens similarly and end up with ?+? = 2?.
Very nice post.
A quibble. I just happen to be brushing up on ordinal arithmetic this week. Ordinal multiplication is defined backwards from our intuition in my opinion. [math]\alpha \times \beta[/math] is defined to be [math]\beta[/math] copies of [math]\alpha[/math] concatenated.
So for example [math]2 \omega[/math] means [math]\omega[/math] copies of 2 strung together. The ordinal 2 represents the order 0, 1. If you line up [math]\omega[/math] of those, you get ... drum roll ... [math]\omega[/math].
On the other hand, [math]\omega 2[/math] is two copies of [math]\omega[/math] side-by-side. You can visualize this as 0, 2, 4,6 , 8,...,1, 3, 5, 7, ... the evens-before-odds order. That's [math]\omega + \omega[/math].
Yes there is. If it is Infinity then it should already contain the 1 you’re attempting to add to it. If it doesn’t contain that 1 being added then it’s not infinity, as it is limited to not containing the 1 you are adding. This means what you are calling ‘infinity’ is not limitless at all and so not worthy of the title.
Seriously, you can even add infinity to infinity. Plenty of cases where that happens in mathematics.
If you have two infinity’s, A & B, then you are saying that in order to add infinity A to infinity B that A does not contain B. Which is to say that both A and B are limited or bounded to A and only A or B and only B
This making two infinity’s then leads to the logical conclusion that it is an indefinite number; an undisclosed amount that is limited to not containing that which you wish to add to it; not an infinite quantity as the mathematitions like to insist.
Please feel free to actually deal with the argument. I’m genuinely interested to hear a counter argument, which you have failed to offer so far.
You don't have an argument.
GreenPhilosophyJune 20, 2018 at 20:34#1896200 likes
Infinity isn't a real number, but it is an extended real number. Infinity can be used to describe infinite things, such as an infinitely sized universe.
By the way, I'm pretty bad at math, so don't take my word for it. I should just stop before I spread false information.
The thing you missed here is the unspoken inference you make. The cardinality of the set of Natural Numbers is not infinity (which is defined as having no limits) as by referring to Natural Numbers you are limiting it to Natural Numbers alone. You are not including anything which is not a Natural Number, it does not include different colours, shapes, texture etc. It is a concept limited to that which is considered a natural number.
How was it unspoken if I literally said the assumption (the the natural numbers are infinite)? That aside, you aren't making sense. That the natural numbers do no, for instance, include the Real Numbers does not entail that the set of Natural Numbers is not infinity. In mathematics, infinity is not (as you claimed) defined as "having no limits". In this case that's especially obvious, because by "limit" you're already sneaking in the assumption of finitude (e.g. the natural numbers are finite, somehow, because the set doesn't include other types of numbers). This argument makes no sense.
You can say that the numbers have no end.. or could go on forever.. or go on indefinitely.. but you cannot refer to them as infinite as you contradict yourself by describing them as such. As they are limited... to Natural Numbers. I am aware that mathematicians are fond of using the word infinite, but I would argue that its an illogical thing to do. As I think I have sufficiently shown.
[quote]No because then you're not talking about the infinite any more.
Consider the following:
1. There are two infinite numbers, A and B
2. A is not B, and B is not A.
3. A is larger than B.
this isn't a description of something without limits. You are specifically saying that A is limited to A and does not include B. And that B is limited to B and does not include A. These are limits.
You can say it has no limits in one specific sense but has limits in others, but then you are not referring to the infinite or to a limitless thing anymore.
You are simply ignoring the definition of infinity that mathematicians use and thereby conclude that it's incoherent because of we assumed your definition we'd get a contradiction. QED, your definition is wrong because it leads to a contradiction. That's ridiculous.
Your argument makes an obvious assumption, namely that all infinite sets are of the same size That's quite literally rejected in mathematics. Infinite sets which are countable, like the natural numbers, have the ability to be put into a one-to-one correspondence with a proper subset of themselves, e.g. we can map all the even numbers onto the set of natural numbers. Uncountably infinite sets (e.g. the reals) cannot do this mapping with the natural numbers, entailing that such sets are larger. Your definition leaves no real ability to use infinity in mathematically useful ways, e.g. Calculus.
You are if you are saying this thing has no limits when it defined within the specific limits of Real or Natural numbers as in the examples you gave. You are therefore saying that this thing is both limited and not limited simultaneously. Which is a contradiction. It cannot be A and ~A.
Incorrect. The natural numbers are the counting numbers, so they do no include the reals. That does not entail the Natural Numbers have a finite *cardinality*, it simply means the set of natural numbers leaves out particular types of numbers. This simply means the set of natural numbers has a particular size of infinity.
It comes down to semantics. Infinity can be considered a quantity in terms of transfinite math - so there are actually many "infinities" (aleph-0 is less than aleph-1; there are "more" real numbers than integers). But it's not a quantity in a sense that it corresponds to anything that exists in the material world.
Reply to MindForged
The existence of an actual infinity (vs a potential infinity) is controversial among philosophers. I'm of the opinion an actual infinity cannot exist. I feel strongest about the impossibility of an infinite past, because that would entail a completed infinity: how could infinitely many days have passed?
Physicists accept the possibility of infinity in space and time simply because there is no known law of nature that rules it out. That doesn't imply the philosophical analysis is wrong, it just means that we don't know of any particular limits.
My opinions are consistent with the dominant opinion among philosophers prior to Cantor's set theory, but that doesn't seem like a very good reason to believe an actual infinity exists in the world.
Reply to Relativist A lot of this, in my estimation, doesn't make sense under scrutiny.
I'm of the opinion an actual infinity cannot exist. I feel strongest about the impossibility of an infinite past, because that would entail a completed infinity: how could infinitely many days have passed?
Um, before every day there is another day. QED. Or to put it more directly, the cardinality of the set of days prior to day "n" can be put into a one-to-one correspondence with the members of the set of natural numbers. Ergo, the number of past days are infinite. I don't know if this is actually true, but there is no logical argument against the *possibility* of it.
However, this wasn't even really what I was suggesting. Between any two moments of time there's another moment. That's what I had in mind. And it's even clearer with the divisibility of space. It's nearly always taken to be a continuum, meaning it would be infinitely divisible.
That doesn't imply the philosophical analysis is wrong, it just means that we don't know of any particular limits
What philosophical analysis? If we are adopting perfectly standard mathematics (or even most non-standard math systems) there is no contradiction whatsoever in supposing the past days are infinite. This will play into a bigger point I make at the end.
My opinions are consistent with the dominant opinion among philosophers prior to Cantor's set theory, but that doesn't seem like a very good reason to believe an actual infinity exists in the world.
I hope it doesn't come across rude, but that just reads as "If you ignore the last 150 years of mathematics most philosophers would agree with me". Well that's... a defense anyone can make to defend their belief in whatever.
Look, my broader issue/point is this. The interplay between our beliefs about the world and the formal tools (maths, logics) is more complex than often made out (i.e. the influence goes both ways). However, generally the idea is that our physics needs math to guide it's conjectures, and our beliefs about the world ought to be in line with the dominant physical theories. If maths has explicated infinity as a coherent, precise concept - and it has - then presumably it becomes irrational to say (as I understand you to be saying) that "Yea yea, there's infinity in mathematics and in physics, but if you try to apply it to real things it entails a contradiction." I just don't get it.
Infinity is not a contradictory concept, so how is it supposed to produce a contradiction if applied to real things? Or is it supposed to be a category mistake? But how does that work? We talk about infinite collections in mathematics all the time, it's central to set theory. That doesn't mean infinite collections (or other infinite whatevers) can exist in our universe, just that you cannot rule them out as incoherent and thus fail to obtain in every possible universe.
3rdClassCitizenAugust 25, 2018 at 07:53#2078360 likes
I'm of the opinion an actual infinity cannot exist.
I believe that beyond time and space, that infinity is only a mathematical construct.
We can not deal with infinite time or space, and nothing else in our world becomes infinite.
We measure the totality of energy by the rate of electrical flow times the amount of time, giving us Kilowatt hours.
If we multiply an infinite universe times the infinitely small, moving point in time in which it exists, perhaps the infinities cancel. This seems like a comfortable, perhaps pathological workaround to the notion that nothing can be infinite.
1. A quantity is a specified amount of something. It has a limit. The infinite is that which has no limits and so cannot be quantified. Therefore, not a quantity as not quantifiable.
I would say that it simply isn't countable or computable. Yet it does in my view quite clearly define a quantity.
The problem is that math starts from counting. And hence we have all the problems with infinity.
3rdClassCitizenAugust 25, 2018 at 21:42#2079490 likes
If different infinities have different values then is it really a definite quantity?
Ammount of real numbers = infinite
Ammount of even real numbers = infinite
Does this mean that infinity divided by 2 equals itself?
Does this not violate the definition of a real number? Seems like there was something on this in "analytical geometry"...
3rdClassCitizenAugust 25, 2018 at 22:37#2079570 likes
Ammount of real numbers = infinite
Ammount of even real numbers = infinite
I meant whole, or natural numbers, not real numbers.
If different infinities have different values then is it really a definite quantity?
Different levels infinities have different sizes. They're size is definite. "Infinity" is not one value.
Amount of real numbers = infinite
Amount of even real numbers = infinite
Does this mean that infinity divided by 2 equals itself?
The real numbers have a cardinality (size) larger than that of the even numbers. The even numbers have the same cardinality as the natural numbers, aleph-null, and so are "countably" infinite. The real numbers cannot be put into a one-to-one correspondence with the natural numbers, being that the reals are a larger infinity. Hence the reals are "uncountably" infinite.
Infinity is a Mathematical fiction and should be applied carefully to the World of Physical Things. For example we can say that there are an Infinite number of Natural Numbers. Natural Numbers are Mathematical concepts. But there can not be an infinitely large Pencil in the Universe. A good old fashioned Pencil is made out of a core of Lead or graphite (lets just say Lead). surrounded by a tube of Wood and then a coat of Paint. Take a point exactly in the center of the Lead and then let the Pencil grow in size to Infinity. You will have a Universe that is completely filled with Lead. You can never get to the Wood no matter how far you travel away from the center point (assuming we can travel through Lead). There will be no Wood or Paint in this Universe. The Pencil will become something less than it was when it becomes Infinite. You can not really have an Infinite Pencil.
Infinity is a Mathematical fiction and should be applied carefully to the World of Physical Things. For example we can say that there are an Infinite number of Natural Numbers. Natural Numbers are Mathematical concepts.
Well, infinity is a very useful mathematical concept then. After all, the number "3" doesn't physically exist either.
SteveKlinkoSeptember 18, 2018 at 10:36#2132190 likes
Infinity is a Mathematical fiction and should be applied carefully to the World of Physical Things. For example we can say that there are an Infinite number of Natural Numbers. Natural Numbers are Mathematical concepts. — SteveKlinkoWell, infinity is a very useful mathematical concept then. After all, the number "3" doesn't physically exist either.
3 things can exist but an infinite amount of things can not.
MindForgedSeptember 18, 2018 at 14:11#2132500 likes
Reply to SteveKlinko I don't see how that's a given. Space is infinitely divisible. Whether or not space counts as a "thing" or not I don't think matters, but it's infinite.
SteveKlinkoSeptember 22, 2018 at 12:32#2141800 likes
?SteveKlinko I don't see how that's a given. Space is infinitely divisible. Whether or not space counts as a "thing" or not I don't think matters, but it's infinite.
Space is definitely a Thing. There can be the 3D Space that we are familiar with, but there can also be 4D Space for example. 4D Space is a whole different Thing than 3D Space. If Space can be different Things then there can be no Space. That would be Absolute Nothingness.
It is not known if Space is Infinite or not. It depends on what the value of a particular constant in Physics is found to be. The current thinking is that our 3D Space is Finite but Unbounded in the sense that it curves back around on itself in some way.
MindForgedSeptember 22, 2018 at 22:12#2142780 likes
If space is a thing, it's not the same as the natural understanding of a thing. That's what I was talking about.
You're other point wasn't what I was talking about. I'm not saying space is infinite in breadth, but it can be infinitely divided without hitting some kind of base unit or boundary point.
SteveKlinkoSeptember 23, 2018 at 10:58#2144410 likes
If space is a thing, it's not the same as the natural understanding of a thing. That's what I was talking about.
You're other point wasn't what I was talking about. I'm not saying space is infinite in breadth, but it can be infinitely divided without hitting some kind of base unit or boundary point.
I wouldn't be so sure that Space can be Infinitely divided. I'll give that a Maybe. It might be subject to the Planck Constant. I don't think you should assume that a Physical Space is the same as a Mathematical Space.
- There is no number X such that X > all other numbers
- Because X+1 > X
Space is discrete that’s why we get paradoxes when we assume it’s continuous (Zeno’s paradoxes).
The Zeno paradox, where to walk any finite distance you have first walk half the distance, then half of the remaining distance etc. is a paradox because Zeno is throwing in a false assumption that you are going across each Half in the same amount of time. This is artificially slowing you down. Of course this would make it impossible to cover the total distance when you consider an Infinite amount of Halves. Zeno forgets that at the same time the Half distances are going to Infinity the number of Halves you are traversing (at constant velocity) is going to an Infinite amount of Halves per second. The Infinite Halves per second and the Infinite number of Halves are compensating Infinities that cancel the paradox.
The way you are solving the paradox uses the undefined quantity ‘infinity’ but I acknowledge there are other ways out of Zeno’s paradoxes other than discrete space.
Still I’d argue for discrete spacetime on the grounds:
- there is no such distance as 1/oo mathematically.
- Imagine a particle moving over a finite period of time. Continuous spacetime would require the particle to have occupied a actually infinite number of states which is nonsensical.
Still even if space is continuous, that would only be a potential infinity rather than actual infinity.
RelativistSeptember 23, 2018 at 16:11#2145110 likes
There is a mathematical relation between the various transfinite numbers, and these relations are analagous to size and quantity, but these are not "quantities" in the exact same sense as the quantities of individual real (or natural) numbers.
SteveKlinkoSeptember 24, 2018 at 11:11#2147340 likes
The way you are solving the paradox uses the undefined quantity ‘infinity’ but I acknowledge there are other ways out of Zeno’s paradoxes other than discrete space.
Still I’d argue for discrete spacetime on the grounds:
- there is no such distance as 1/oo mathematically.
- Imagine a particle moving over a finite period of time. Continuous spacetime would require the particle to have occupied a actually infinite number of states which is nonsensical.
Still even if space is continuous, that would only be a potential infinity rather than actual infinity.
I say that Infinity is a Mathematical Fiction that only exists in the world of Mathematics. But since the construction of the Zeno Paradox uses Infinity as the basis of the argument we must accept the premise and argue from that. I think the compensating Infinity argument is the best way out of the Paradox.
If a particle only occupied discrete states then according to your theory it would have to jump from position to position while moving. It would necessarily have to stop at each position for the time it would take to continuously travel between two of the positions. This is as nonsensical as a continuous movement with Infinite intermediate positions. These are both nonsensical and serve to illustrate the problems you can get into when you think a little more Deeply about things. Good Thoughts however.
If a particle only occupied discrete states then according to your theory it would have to jump from position to position while moving. It would necessarily have to stop at each position for the time it would take to continuously travel between two of the positions.
A good point. It depends on your view of time as to whether you think the particle exists in an actually infinite number of states:
- Presentist. The past does not exist. So the particle does not exist in an Actually Infinite number of states, just one state, the present.
- Eternalist. The past exists so continuous time implies the particle must exist in an actually infinite number of states.
Presentism leads to paradoxes, so that suggests Eternalism. But time must be discrete for Eternalism to be free of Actual Infinity (which I class a paradox).
SteveKlinkoSeptember 25, 2018 at 11:07#2149880 likes
If a particle only occupied discrete states then according to your theory it would have to jump from position to position while moving. It would necessarily have to stop at each position for the time it would take to continuously travel between two of the positions. — SteveKlinko
A good point. It depends on your view of time as to whether you think the particle exists in an actually infinite number of states:
- Presentist. The past does not exist. So the particle does not exist in an Actually Infinite number of states, just one state, the present.
- Eternalist. The past exists so continuous time implies the particle must exist in an actually infinite number of states.
Presentism leads to paradoxes, so that suggests Eternalism. But time must be discrete for Eternalism to be free of Actual Infinity (which I class a paradox).
The Infinitely Large and the Infinitely Small are a real pain in the Brain.
SteveKlinkoSeptember 26, 2018 at 11:24#2154200 likes
Imagine a Square drawn on a piece of paper. Now imagine the Square shrinking smaller and smaller. It remains a Square no matter how small it shrinks. If we stop shrinking it and start magnifying it back we can bring the Square back to the original size. But now imagine the Square shrinking to Zero size. All points of the Square collapse to a single point and there is no longer a Square on the paper. The square has been transformed into a single point. We would not be able to magnify the resulting point back the the original Square. We could also shrink a Triangle in the same way and at Zero size it would be a single point just like the Square. The Square and the Triangle lose their identity when they are Zero size. They become something different. They become something less than what they were. Zero size is an unrecoverable threshold of size that changes everything.
Now imagine a Square that is the smallest Square that is not equal to Zero. This thought sends your mind into an endless recursive loop of the Square getting smaller and smaller and we soon realize that it is impossible to imagine such a smallest Square. One thing we can say is that this Square is Infinitely small but is still a Square. In general mathematics this would be called a differential Square or an infinitesimal Square.
Next imagine the Square that was drawn on the paper growing larger and larger. If the Square was exactly in the center of the paper the sides of the Square would eventually move off of the paper and past the edges of the universe. It remains a Square no matter how large it grows. If we stop growing it and start shrinking it back we can bring the Square back to the original size. But now imagine the Square growing to Infinite size. The sides would all move out to infinity. No matter how far you went in the universe you would never encounter a side of the Square. The Square has effectively exited the universe. We could also grow a Triangle in the same way and at Infinite size it will no longer be found in the universe. The Square and the Triangle lose their identity when they are Infinite size. They become something different. Paradoxically they become something less than what they were. You might think that the Square and Triangle are still out there at Infinity. But there is no "there" at Infinity. The Square and Triangle are gone. If you think you can go out "there" to an edge of the Square or Triangle at Infinity then that "there" is not Infinity. Infinite size is an unrecoverable threshold of size that changes everything.
Now imagine a Square that is the largest Square that is not equal to Infinity. Similar to the differential Square, this thought sends your mind into an endless recursive loop of the Square getting larger and larger and we again soon realize that it is impossible to imagine such a largest Square. We can say that this Square is Infinitely large but is still a Square that exists in the universe.
I think that just as Infinite Squares are not possible it is probably true that any Infinite Physical quantity of anything is not possible. Just because an equation in Science goes to Infinity, it doesn't mean that the Physical quantity in the equation is able go to Infinity. I think this is a limitation of what we can do with Mathematics. Seems like a minor limitation but it has big consequences when equations in Science go to Infinity.
But now imagine the Square growing to Infinite size. The sides would all move out to infinity. No matter how far you went in the universe you would never encounter a side of the Square.
A good point. Both the process of growing something to oo and shrinking something to 1/oo are destroying information, which is not meant to happen in the physical world.
A line is an infinite quality as 1 direction with this infinite direction as 1 being composed of multiple line as 1 infinity in themselves.
Hence infinity, through the line as one considering one premised in empirical qualities observes all numbers as an observation of linear qualities through time as directive in nature.
The line as 1 is infinity as 1 with one itself being infinite in the respect it is continuous as existing through itself as itself. We can observe that a line as 1 through 0d space or void must effectively projects if it is to exist. However considering the line has nowhere to project but must project to something if it is to maintain its nature, the line must effectively fold through itself by multiplying in directions.
In these respects the 1 original line as infinite must individuals into multiple lines which each line being a ratio relative to each other and the original line while being infinite and one in itself. Finiteness is strictly the relation of infinities with infinity, as an absence of finiteness, being an absence of relation with infinity and finiteness existing at the same time in different respects due to there positive and negative qualities.
Not only is infinity quantifiable but there is one infinity and multiple infinities with these multiple infinities being the premise for finiteness or time.
In simpler terms the 1 directional nature of the infinite line shows infinity it only as quantifiable but effectively existing as a limit.
Limitless ScienceOctober 15, 2018 at 21:59#2206240 likes
No, limitlessness is. However you can't really take it as one because it's limitless. Infinity in the other hand is a process of something that takes forever. Limitlessness is the ultimate/perfect word. However it is only possible if there's no beginning and no end.
Is infinity properly thought of as a number? Is it a quantity? Is that the same question?
Is a bishop a religious guy with a cool hat? Or is a bishop a piece in a certain game that moves diagonally? There are connections between the hat-guy and the chess-piece, but they are different, and they both make pretty good sense in their context.
In the first case, infinity is a shorthand for a limiting process (the infinity is hidden in the quantifier 'for all epsilon')
Since you said correctly that in the definition of limit the notion of infinity is hidden in the quantifier I think you are not confusing the limit to infinity (infinity indicates the graphical correspondent to the behavior of a function, in so far as its values become arbitrarily large) with infinity of the limit(infinity limit a process) I suggest to not use the misleading terms: 'limiting process' but 'unlimited variables bounded in regards to which the limit is considered'.
Of course tom is right(but unfortunately unsuccessful) in preventing the tedious error of conceiving of infinity as a set of numbers. More explicitly: infinity is a relational concept, and its use as a factor just means the operations operate on variables(and infinite variables[of numbers] plus the unity is of course a not problematic thing). One thing is infinity as relational concept, another is the Symbol of infinity to indicate infinitely many variables.
In the second case - for cardinals - they give the size of infinite sets, so yes they are probably quantities since they represent the magnitude of something.
I don't think cardinality offer a quantitative view of Infinity, since it is either a relation between a set and its elements or between its elements and numbers(e.g. a set is D-
infinite iff for every natural number the set has a subset whose cardinality is that natural number) or between sets(e.g. the cardinal of R is bigger than the cardinal of I)
SophistiCatNovember 30, 2018 at 08:28#2324230 likes
I don't think cardinality offer a quantitative view of Infinity, since it is either a relation between a set and its elements or between its elements and numbers(e.g. a set is D-
infinite iff for every natural number the set has a subset whose cardinality is that natural number) or between sets(e.g. the cardinal of R is bigger than the cardinal of I)
Numbers are sets, in the usual axiomatizations, so cardinality very naturally fits our idea of quantity.
Metaphysician UndercoverNovember 30, 2018 at 11:43#2324340 likes
Numbers are sets, in the usual axiomatizations, so cardinality very naturally fits our idea of quantity.
The only problem here is that "sets" are based in qualities, and there is a conceptual difference between quality and quantity. Therefore set theory does not naturally fit our idea of quantity. So set theory provides a set of axioms which modify mathematics in a way so as to be inconsistent with our natural idea of quantity.
The only problem here is that "sets" are based in qualities,
This is false. Sets are based on RELATIONS(between something, i.e. a set, and its elements. The empty sets have a relation such that no elements belongs to it).
Therefore set theory does not naturally fit our idea of quantity. So set theory provides a set of axioms which modify mathematics in a way so as to be inconsistent with our natural idea of quantity.
These are YOURS(wrong) assertions or opinions about what intuitive quantity should be and how this relates to set theory and about set theory.
This is false. Sets are based on RELATIONS(between something, i.e. a set, and its elements. The empty sets have a relation such that no elements belongs to it).
A relation is a comparison between a quality of one, and a quality of another. Therefore it is inherently qualitative.
Define explicitly a quality without presuming a quantity man, if you can. You will find you can not, and since quantity presupposes a relation, insofar as a quantity is relation between spaces, then quality presupposes relation.
I add: YOU think relations are comparison, but this is a very special case of relations. I'll give you a fair example of a relation which is not a comparison: a function. You set your prejudices as metaphysical truth, and this just because your ignorance of the advancements or acknowledgements of mathematics and logic.
Metaphysician UndercoverNovember 30, 2018 at 12:09#2324430 likes
Reply to Ikolos
Quality is the degree of excellence of a thing. This implies a relation, and as I said, relations are inherently qualitative. However, it does not imply a quantity, as quantity requires discrete units, and "degree" may be applied to a continuum. It is only when we refer to individual "degrees" that quantity is implied in such a relation.
Metaphysician UndercoverNovember 30, 2018 at 12:21#2324450 likes
As you can see, we can quantify a quality and this is what provides us with the capacity to measure. But to qualify quantity, and this is what set theory does, is a mistaken procedure because it inhibits our capacity to measure, by imposing qualitative restrictions on quantity.
SophistiCatNovember 30, 2018 at 12:27#2324480 likes
MindForgedNovember 30, 2018 at 15:06#2324600 likes
Reply to Ikolos I really don't know what you're talking about. Infinity (as in the cardinality and ordinality of sets which can be put into a function with a proper subset of themselves) is a quantity. Finite Numbers can be defined in terms of the cardinality of sets in just the same way.
Finite Numbers can be defined in terms of the cardinality of sets in just the same way.
This is not even wrong. It has no sense. Numbers are not defined otherwise than by postulating a number and stating which operations preserves that being a number. See basic, high school Math(Peano Axiomatization of Arithmetics).
Frankly, a definition so senseless as all the medieval definitions were. You are deeply confused. You can quantify on something, by selecting somehow a unit of quantity, only if there is yet a quantity independently from that selection.
Furthermore it does not give us the 'capacity of measuring', because, in fact, a measure(as you may intend, for the use of terms by you is ambiguous: physical measure? or which one?) presupposes a mode of measuring, and so presupposes a quantity on which that method is to be applied. In physics, this is matter, or whatever you may call it.
But since you are so darkened on thoughts about what a quality is I will give you a fair dispute, stating better your naive argument and than responding to this peter version(be honored: the argument is by Kant):
«Reality in appearances(i.e. matter as we perceive it) always has a magnitude(a scale of degrees, i.e. quality), which is not, however, encountered in apprehension(you do not apprehend the space in which it is by itself) , as this takes place by means of the mere sensation in an instant and not through successive synthesis of many sensations, and thus does not proceed from the parts to the whole; it therefore has a magnitude, but not an extensive one.» p.290 of Critique of pure reason (guyer wood)
Then he continue:
«Now I call that magnitude which can only be apprehended as a unity, and in which multiplicity can only be represented through approximation to negation = 0, intensive magnitude. Thus every reality in the appearance has intensive magnitude, i.e., a degree. » ibid.
Now, if something is to be selected as a discrete unity, as you say, it presupposes a quantity on which this selection is operated. Or do you think we actually create quantity by itself, against the basic postulate of physics?
Kant is saying that we do not apprehend it, in the sense that the homogeneity of the synthesis is a presupposition, and also it renders possible at all to select a unity: for a unity is a unity of a manifold, and if there were no homogeneity it would not be the unity of that manifold.
It is only deniable by pathologically affected(in the brain) people that homogeneity is a spatial property. And quantity in general if we have to be cautious, thus not saying quantity in general is space(which is a big assumption), we say that PRESUPPOSES ONLY SPACE, as its parts are external to one another: this sufficing to provide an account of quantity, on the postulate(common to every physical theory that has any sense) that space is occupied by something(I,e. matter), and THUS a quantity is possible at all, insofar as we can distinguish it from the space itself and thus it is possible to be measured with a certain referential unity.
Hence as the scale of degrees(quality) relies on a spatial property, than the degrees do rely on that to. But that property alone can not give us any quantity, for, by a quantity, we do not intend a mere externality between parts, but an OCCUPATION of (parts of) space. Hence Space(spatial properties we are able to detect) and matter are presupposed by any concept of quality that has any sense.
But neither Space nor matter presupposes any detectable quality by themselves. Quality, furthermore, presupposes a RELATION between space and something, which renders possible to detect some spatial properties or, as you seem to prefer, to select from that properties units, in respect to which establish a scale of measuring. This something is matter. Matter does not imply quality(degrees) but the distinguishability of degrees implies matter. But matter presupposes space. Then quality presupposes space. Either you identify space with the properties we can distinguish and classify under the kind 'spatial' and name IT quantity, or you do not identify space with those properties and call those properties 'QUANTITY' it is the same for our question: quantity it is presupposed by quality.
MindForgedNovember 30, 2018 at 17:07#2324720 likes
This is the 'singoletto'(I don't know the English word, kind of 'single set') not of a number.
The word you're thinking of is "singleton" I believe. And no, that's not just the singleton of a set. It was the definition of the number one in set theory. If Zero is defined as the empty set, One can be defined as it's successor; the union of Zero with singleton-Zero.
I'm still confused about what you meant when you said I had a medieval theologian type of thinking...
This is the condition to CONSIDER the 'singleton'(I trust you on the term) as the number one, but not to DEFINE it. Numbers are defined by a postulate and succession: see Peano Arithmetics( 0 is number(POSTULATE). If n is a number, then n+1 is number.(AXIOM) Then 1 is a number, as 0+1=1.)
I'm still confused about what you meant when you said I had a medieval theologian type of thinking...
Not referring to you in particular. By 'medieval thinking of infinity'(and conception more or less related to it) I mean a very precise thing that I say to you.
Thinking infinity has THE QUANTITY OF WHICH NO QUANTITY IS GREATER is the MEDIEVAL conception of infinity.
Since Kant gave (1781!!) an explanation on why this is wrong, I cite from him(Guyer Wood translation, p.472)
«I could also have given a plausibleb proof of the thesis by presuppos ing a defective concept of the infinity of a given magnitude, according to the custom of the dogmatists. A magnitude is infinite if none greater than it (i.e., greater than the multiplec of a given unit contained in it) is possible.58 Now no multiplicity is the greatest, because one or more units can always be added to it. Therefore an infinite given magnitude, and hence also an infinite world (regarding either the past series or ex tension), is impossible; thus the world is bounded in both respects. I could have carried out my proof in this way: only this concept does not agree with what is usually understood by an infinite whole. It does not represent how great it is, hence this concept is not the concept of a maximum; rather, it thinks only of the relation to an arbitrarily as sumed unit, in respect of which it is greater than any number. According as the unit is assumed to be greater or smaller, this infinity would be greater or smaller; yet infinity, since it consists merely in the relation to this given unit, would always remain the same, even though in this way the absolute magnitude of the whole would obviously not be cognized at all, which is not here at issue.»
I will, a time or another, explain in details how mathematical infinity relates to what Kant meant by Transcendental CONCEPT of infinity, i.e. the relation between math-logical and epistemological infinity.
SophistiCatNovember 30, 2018 at 17:24#2324740 likes
Reply to MindForged Our cranky and inarticulate friend has a point in that there is a difference between a conceptual definition of a number, which describes the properties that anything fitting the definition of a 'number' ought to have, and its particular theoretical construction, such as von Neumann's (which was designed to meet the requirements of the conceptual definition).
But my comment about numbers being sets (everything is a set in the set theory construction of mathematics - obviously) was made in the context of the preceding discussion, which Ikolos does not or will not follow.
MindForgedNovember 30, 2018 at 17:58#2324840 likes
This is the condition to CONSIDER the 'singleton'(I trust you on the term) as the number one, but not to DEFINE it. Numbers are defined by a postulate and succession: see Peano Arithmetics( 0 is number(POSTULATE). If n is a number, then n+1 is number.(AXIOM) Then 1 is a number, as 0+1=1.)
Thinking infinity has THE QUANTITY OF WHICH NO QUANTITY IS GREATER is the MEDIEVAL conception of infinity.
Well I that's definitely not how I described infinity, and it doesn't even seem correct because it's too vague. As there are many sizes of infinity, I don't even think I could pretend to accept "the quantity of which no quantity is greater" as a description of infinity.
MindForgedNovember 30, 2018 at 18:01#2324850 likes
?MindForged Our cranky and inarticulate friend has a point in that there is a difference between a conceptual definition of a number, which describes the properties that anything fitting the definition of a 'number' ought to have, and its particular theoretical construction, such as von Neumann's (which was designed to meet the requirements of the conceptual definition).
Hm, okay but I don't really get what their point about infinity was supposed to be. It didn't really sound like something which would accurately characterize how the concept is used and understood in modern math. Aleph-null is definitely not a quantity above all others, for instance.
Metaphysician UndercoverNovember 30, 2018 at 19:54#2325080 likes
Frankly, a definition so senseless as all the medieval definitions were. You are deeply confused.
It's the #1 definition in my OED under "quality". I really think that it's you who is confused. would you prefer:"a distinctive attribute or faculty: a characteristic trait"?
Now, if something is to be selected as a discrete unity, as you say, it presupposes a quantity on which this selection is operated. Or do you think we actually create quantity by itself, against the basic postulate of physics?
You clearly have this backward. Unity does not presuppose quantity. Quantity follows from unity, that's how we measure, by units. Quantity requires the individuation of units, therefore it requires unity. Unity does not presuppose quantity. So it s quite clear that we do create quantity, as it is minds that individuate units such that something may be counted. Which basic postulate of physics states otherwise?
Hence as the scale of degrees(quality) relies on a spatial property, than the degrees do rely on that to.
Yes, I agree that "degrees" rely on "quantity". And a difference of quality is measured as a difference of degree, that's the point I made. This is what allows us to measure different qualities. When we notice difference, and we produce "degrees" to measure that difference, we produce the means to quantify that quality. The units of measure, "degrees", by which the qualitative difference is divided, may be a completely arbitrary production of the mind. And, this arbitrary division of the qualitative difference, into fundamental units, "degrees", may give us the capacity to measure the quality.
But neither Space nor matter presupposes any detectable quality by themselves. Quality, furthermore, presupposes a RELATION between space and something, which renders possible to detect some spatial properties or, as you seem to prefer, to select from that properties units, in respect to which establish a scale of measuring. This something is matter. Matter does not imply quality(degrees) but the distinguishability of degrees implies matter. But matter presupposes space. Then quality presupposes space. Either you identify space with the properties we can distinguish and classify under the kind 'spatial' and name IT quantity, or you do not identify space with those properties and call those properties 'QUANTITY' it is the same for our question: quantity it is presupposed by quality.
You are misrepresenting the difference which exists within the thing, matter, and replacing it with our measurement of that difference, which is with degrees. Degrees of difference are not necessarily distinguishable because "the degree" may be arbitrary. The premise of "the distinguishability of degrees" misleads you. So we commonly divide space by degree with no implication of matter.
SophistiCatNovember 30, 2018 at 20:53#2325220 likes
Reply to MindForged Yeah, his communication skills leave much to be desired, in more ways than one. I am not going to bother with him any further.
everything is a set in the set theory construction of mathematics - obviously)
This is naive set theory darling. Thank you for your opinions directly expressed on my communication skills. I am sorry you don't understand set theory, but that's not my fault.
Yes, and what do you measure my friend? Quantity. Hence there is quantity yet, and you measure it. That fact that the methods and units of measurement may be relative(as you correctly say) does not make this less true, i.e. quantity is presupposed REALITER as what is to be measured.
"the degree" may be arbitrary. The premise of "the distinguishability of degrees" misleads you. So we commonly divide space by degree
The degree as a reference of measure is arbitrary, but clearly the degree of complexity of bacteria is inferior to that of an dog as an organism. No one divide space by degree man, that is nonsensical. Topology does not consider differences of space by degrees at all.
The differences in the things, as you say, is a difference which is not merely spatial, but involve matter, hence degrees. Because if you can not recognize the difference(with a possible measure) things are not at all distinguishable, except by logical or topological characteristics. Unfortunately, we are sensible beings.
Is different in a very significant way. Because CONSIDER something as something else(with reasons to do that) is a thing, DEFINE(as you said) something is another. Citing you:
that's not just the singleton of a set. It was the DEFINITION of the number one in set theory.
Numbers are not defined in set theory. They are considered elements of sets. Numbers are defined by induction in Peano Arithmetics, postulating 0 as a number and posing the axiom of succession.
Metaphysician UndercoverDecember 01, 2018 at 12:34#2326280 likes
Yes, and what do you measure my friend? Quantity. Hence there is quantity yet, and you measure it. That fact that the methods and units of measurement may be relative(as you correctly say) does not make this less true, i.e. quantity is presupposed REALITER as what is to be measured.
No, this is clearly not the case. The units of measurement, which are counted as a quantity, are very often created by the human mind, for the purpose of measurement, they are artificial. So "quantity", as the thing measured, is not necessarily presupposed. In the cases where we cannot find individual units to count, we simply create them, giving us the capacity to measure without there being any real quantity which is being measured.
We're talking about measuring quality, which is said to differ by degree, and the problem is that the units of degree are often artificial and may be arbitrary. For instance there are degrees of temperature. The thing measured is a difference, the unit of measurement, one degree, is artificial and arbitrary. So the quantity is artificial, and there is no actual quantity which is being measured, the "quantity", as the measurement, is assigned to it, arbitrarily, it is not something within the thing measured.
You can also look at the 360 degrees of a circle in the same way. That number of degrees around a circle is complete arbitrary. The circle could be divided into an infinite number of degrees, and there could be an infinite number of degrees between each of the four right angles, and any other angle around the circle, expressing a continuity around the circle, rather than the discrete units of degree which are commonly used. This is actually expressed with minutes and seconds. There is no real quantity of measurable units which are being measured around the circle.
Infinity is a symbol given to an unlimited amount of things.
Btw, countable and uncountable infinity is counterintuitive, insane and nonsensical. Cantor was a lunatic.
In the diagonal proof you have an infinite amount of rows and columns. Supposedly you can make a new infinite number different from the one on the list. How? By taking the first digit and changing it to whatever, then the second digit, and so on. This is nonsense.
First of all the new number has conditioned itself to be infinitely different from the one on the list. In no possible way can the number be the one on the list, however the same can be said by the number on the list ever actually being a quantity. It has and will never end. Thus they're both infinite numbers, and thus they both have the same length. The only difference is the rate at which they grow.
Second of all the rate of growth in infinite numbers is not a valid argument to define N (aleph-null) as an ordinal 'uncountable infinite'. This is because the rate of growth can still be achieved without ever actually having the list to being with. This new number N is in no way different or special from the numbers on the list.
Third of all the rate of growth to infinity does not change at all the fact that they're both infinite. Saying that some N1 is infinite will then have an N2 being infinite proceed from that one makes no sense and is mathematically possible but absolutely irrational, and a waste of time for your brain to acknowledge. But people don't understand these three simple concepts.
The units of measurement, which are counted as a quantity, are very often created by the human mind, for the purpose of measurement, they are artificial. So "quantity", as the thing measured, is not necessarily presupposed. In the cases where we cannot find individual units to count, we simply create them, giving us the capacity to measure without there being any real quantity which is being measured.
Planck Length, Planck Scale, speed of light (which is basically a scale constant) are not in any sense arbitrary. Unless Im mistaken, the SI system is based on non-arbitrary physical constants not this nonsensical notion of "qualities" or whatever. And even if it isn't, such a thing is possible but the gain is little for all the work required. See Natural units.
That number of degrees around a circle is complete arbitrary. The circle could be divided into an infinite number of degrees, and there could be an infinite number of degrees between each of the four right angles,
That something could be done a different way does not make something arbitrary. There are perfectly sensible reasons to put the number of degrees at 360. It's a highly composite number allowing us to avoid fractions (which are hard for humans to do, hence the preference for decimal expressions), it's not a large whole number so it's fairly easy to do basic math with (particularly division), etc. I'm thinking you're using a weird definition of "arbitrary" or not explaining why it is (supposedly) so.
MindForgedDecember 02, 2018 at 03:25#2327920 likes
#1 is false in some sense. Unless you ignore modern math, there are a hierarchy of infinities. #2 is extremely wrong, there are many infinite numbers. The transfinite cardinals and ordinals will fit any sensible, non-arbitrary definition of a number.
First of all the new number has conditioned itself to be infinitely different from the one on the list. In no possible way can the number be the one on the list, however the same can be said by the number on the list ever actually being a quantity. It has and will never end. Thus they're both infinite numbers, and thus they both have the same length. The only difference is the rate at which they grow.
Who cares if it's "conditioned"? That is not a criticism, you're whining with this non-objection. The mere fact (proven by the opposite entailing a contradiction) that such a number necessarily exists is why mathematicians accept Cantor's diagonal argument. "Never actually being a quantity" is, naturally, something you do not demonstrate to be the case. That the number "never ends" does not mean they have the same length, that's a non sequitur. Show exactly where the "conditioned" number will appear in the original set. That you cannot (again, because it would be a contradiction) entails the different sizes of the respective sets.
Second of all the rate of growth in infinite numbers is not a valid argument to define N (aleph-null) as an ordinal 'uncountable infinite'. This is because the rate of growth can still be achieved without ever actually having the list to being with. This new number N is in no way different or special from the numbers on the list.
Aleph-null is not an ordinal, it's a transfinite cardinal. Literally the smallest, first transfinite cardinal is aleph-null. And thus it is not uncountably infinite, it's countably infinite, because the definition of that is "being capable of being put into a function with the set of natural numbers".
Third of all the rate of growth to infinity does not change at all the fact that they're both infinite. Saying that some N1 is infinite will then have an N2 being infinite proceed from that one makes no sense and is mathematically possible but absolutely irrational, and a waste of time for your brain to acknowledge. But people don't understand these three simple concepts.
"Both infinite" says nothing more than that they're both the same type of number. That "0.33" and "0.44" are both real numbers doesn't mean they're both the same number. You are under a terrible misapprehension. If the sets were the same size they could be mapped onto each other. That's how size is defined in math, they have to have the same cardinality. But of course, if you cannot even accept Cantor's proof by contradiction that the naturals and the reals cannot be the same size then I'm not surprised you fail to grasp this.
Cut the nonsense. Show exactly where the sets map together to make them the same size. That you cannot means they aren't the same size. Ergo, one is larger and the other smaller. It's funny you say it's mathematically possible yet "absolutely irrational". I'm sure you can defend that characterization as coherent. One would assume mathematics cannot be incoherent, given incoherency in mathematics means triviality; everything becomes a theorem if the math is incoherent, but there's absolutely no evidence that current mathematics is trivial. Show the formal contradiction. You'll win a Fields Medal.
Metaphysician UndercoverDecember 02, 2018 at 04:01#2327960 likes
Planck Length, Planck Scale, speed of light (which is basically a scale constant) are not in any sense arbitrary. Unless Im mistaken, the SI system is based on non-arbitrary physical constants not this nonsensical notion of "qualities" or whatever. And even if it isn't, such a thing is possible but the gain is little for all the work required. See Natural units.
As I said, and provided examples for, some divisions of degree are arbitrary, I didn't say that all are arbitrary. But the fact that some are, is all that's required to disprove Ikolos' claim that quantity is what is measured. Actually, quantity is the measurement.
That something could be done a different way does not make something arbitrary. There are perfectly sensible reasons to put the number of degrees at 360. It's a highly composite number allowing us to avoid fractions (which are hard for humans to do, hence the preference for decimal expressions), it's not a large whole number so it's fairly easy to do basic math with (particularly division), etc. I'm thinking you're using a weird definition of "arbitrary" or not explaining why it is (supposedly) so.
That's not even an argument, the number of degrees in a circle is not arbitrary, it was chosen because it's "easy to do basic math with". The principle divisions of the circle are the four right angles. So the number of degrees in a circle need only be divisible by 4 in order that "it's fairly easy to do basic math with". That the circle consists of 360 degrees, and not 4, 8, 12, 16, 20, 400, 800, or any other number of degrees divisible by 4 is arbitrary. That the number of degrees in the right angle is 90, and not 1, 2, 3, 4, 5, 100, 200, or any other number, is completely arbitrary.
MindForgedDecember 02, 2018 at 04:52#2328020 likes
As I said, and provided examples for, some divisions of degree are arbitrary, I didn't say that all are arbitrary. But the fact that some are, is all that's required to disprove Ikolos' claim that quantity is what is measured. Actually, quantity is the measurement.
Incorrect, their argument was that some were not "qualities" as you deemed them because they are part of reality. Pointing out that some aren't (as that user already admitted) is very much besides the point when they already admitted so.
That's not even an argument, the number of degrees in a circle is not arbitrary, it was chosen because it's "easy to do basic math with".
How is that not an argument? Ease of use is a perfectly legitimate reason to do prefer something. You user particular kinds of screw heads in because they're easier to use for particular kind of screws (this generalizes to tools in general). Hell, the entire reason a radian is defined as 2 x Pi is because it makes calculus and trigonometry easier if angles are measured in radians instead. Again, you're working under a strange definition of "arbitrary" if practicality doesn't count because it's very helpful that a circle has 360 degrees. Also, try to do set a circle equal to 4 degrees and see how the math works out for you. Other numbers have many divisors, but they aren't as useful mathematically (example, 2570 can be cleanly divided by every number 1 - 10 but it's not very useful in geometrical calculations to set it that way).
That's a very good damn response.
Logically I would state that the reason why aleph-null makes no sense is because like both aleph-null and the numbers on the list are infinity, then to say that aleph-null is in any way different from the numbers on the list is to state that the numbers on the list have an ending. This is because for them to actually be different is for both numbers to end in a difference. Such statement may sound illogical at first glance because no matter what number appears it will always be different, however the point is that if we think about it aleph-null is still impossible to fully become different.
All the notation that comes after aleph-null is to assume the difference. Here is a mathematical intepretation of my reasoning.
j = {0, 1, 2, 3, 4 ... M}
N = {1, 2, 3, 4, 5 ... X}
j = numbers on the list
N = Aleph-null
X= Statement of difference
M = hyperreal of j
however
N = {1, 2, 3, 4, 5 ... X },
N = {1, 2, 3, 4, 5 ... M1 ... X ... },
N = {1, 2, 3, 4, 5 ... M1 ... M2 ... X ... },
To say that aleph N is different from j is to say that j ends at some M for it to become X. It doesn't and never will become X (different).
Even mathematically, to state that aleph-null is a thing is to reduce the nature of the problem to cardinality. Which is meaningless. The infinity to be reasoned in any physical sense is to acknowledge that impossibility of convergence to some difference.
Also, the reals are bigger than natural numbers because they're defined in terms of themselves. They are not conditioned to exist to something which will never end. In this case aleph-null is conditioned to exist for when M ever becomes X - It will never do such a thing.
That's a very good response.
Logically I would state that the reason why aleph-null makes no sense is because like both aleph-null and the numbers on the list are infinity, then to say that aleph-null is in any way different from the numbers on the list is to state that the numbers on the list have an ending. This is because for them to actually be different is for both numbers to end in a difference. Such statement may sound illogical at first glance because no matter what number appears it will always be different, however the point is that if we think about it aleph-null is still impossible to fully become different.
All the notation that comes after aleph-null is to assume the difference. Here is a mathematical intepretation of my reasoning.
j = {0, 1, 2, 3, 4 ... M }
N = {1, 2, 3, 4, 5 ... X }
j = numbers on the list
M = hyperreal of j
N = Aleph-null
X= Statement of difference
however
N = {1, 2, 3, 4, 5 ... X },
N = {1, 2, 3, 4, 5 ... M ... X2... },
N = {1, 2, 3, 4, 5 ... M ... M2 ... X3 ...},
To say that aleph is different from j is to say that j ends at some M for it to become X. It will never do such a thing.
Now, the real numbers are greater than the naturals because they're both defined by themselves to be so. In this case aleph is defined to exist as long as it satisfies the difference it has with j. It doesn't have one because it cannot happen due to the nature of what is infinity.
Mathematically you can reduce all this into cardinality but it throws out the window the sense of infinity in any physical and reasonable sense, and so it defines things in a finite way similar to a hyperreal. Stating that it exist if one day M stops growing to become X.
Even then the argument behind a hyperreal is more logical then that of aleph-null. Because the hyperreal is taken into account to never be n. And so, to be able to stay at some n.
The difference X involves taking some n of j to be x. Which is to state that M is an n. No logos my buddies, what the heck is going on?
With this I say that these mathematics are physically inconsistent. So you'll never see anything like this in the physical world. To think this is amazing is to be amazed at 2 + 2 = 3 but in a more complex and sophisticated way
MindForgedDecember 02, 2018 at 06:58#2328610 likes
Logically I would state that the reason why aleph-null makes no sense is because like both aleph-null and the numbers on the list are infinity, then to say that aleph-null is in any way different from the numbers on the list is to state that the numbers on the list have an ending. This is because for them to actually be different is for both numbers to end in a difference. Such statement may sound illogical at first glance because no matter what number appears it will always be different, however the point is that if we think about it aleph-null is still impossible to fully become different.
That does not follow. For one to be larger than the other all that need be true is that one set has a greater cardinality. What this will mean is that when you try to place them in a one-to-one correspondence with each other, it fails to be possible to do so. After all, sets that can be mapped together in this way are the same size. What Cantor showed was that it's impossible to map the naturals with the reals on pain of contradiction, it turned out the reals were larger not that the naturals had an end (in the sense of a final member). That's what makes them different, despite being infinite. They're different levels of infinity.
That does not follow. For one to be larger than the other all that need be true is that one set has a greater cardinality. What this will mean is that when you try to place them in a one-to-one correspondence with each other, it fails to be possible to do so. After all, sets that can be mapped together in this way are the same size. What Cantor showed was that it's impossible to map the naturals with the reals on pain of contradiction, it turned out the reals were larger not that the naturals had an end (in the sense of a final member). That's what makes them different, despite being infinite. They're different levels of infinity.
I get the point. The ending member is different in a one to one correspondance from T (numbers on the list) to N (aleph). I would just like to believe that from R to N (naturals) the last member is different because that's how they are defined by themselves. If we make a number out of the difference of a specific number is to state that the former was a number at all, even in regards to the cardinality of a set.
I would like to believe that this up here is sufficient to say that aleph as a set was to just be math dribble. And the last member of the set T never existed in order for it to be different. So the definition of the set aleph is not satisfied.
Metaphysician UndercoverDecember 02, 2018 at 14:37#2328920 likes
Incorrect, their argument was that some were not "qualities" as you deemed them because they are part of reality. Pointing out that some aren't (as that user already admitted) is very much besides the point when they already admitted so.
I can see that one, or both of us, misunderstands what Ikolos was saying. I can restate what I was saying though. I said that qualities can be quantified, but it is a mistake to attempt to qualify quantities. Both qualities and quantities are "part of reality", so this reference is just a diversion. The issue is what the mind is doing when it attempts to quantify a quality, or qualify a quantity.
Ikolos replaced "quality" with "relation", and I had to insist that relation is a quality rather than a quantity, because Ikolos wanted to argue that a relation is a quantity, without the required mental act which quantifies that quality.
Also, try to do set a circle equal to 4 degrees and see how the math works out for you.
If a circle were 4 degrees, I see that an acute angle would be less than one degree, and an obtuse angle would be greater than one degree. Forty five would be half a degree, and one eighty would be two degrees. Looks very easy to me. I'm no mathematician so I might be missing something. Where would the mathematical problem be, which would make 360 degrees mathematically easier than 4 degrees?
Comments (87)
There's what the usual infinity symbol represents: [math]\infty[/math], which usually denotes a limiting process: [math]\lim_{x\rightarrow\infty}f(x)[/math], IE what value [math]f(x)[/math] tends to when [math]x[/math] becomes arbitrarily large. Formally this corresponds to a definition of a limit and can be considered shorthand for it.
Then you've got cardinal numbers, which count how many of something there are. The smallest infinite cardinal is called [math]\aleph_0[/math], which is the size of the set of natural numbers [math]\mathbb{N}=\{0,1,2,3,...\}[/math]. Then there are ordinal numbers, which agree with cardinal numbers up to [math]\aleph_0[/math] and can disagree beyond that - they correspond to different ways of ordering infinite sets of things. For example, the standard ordering of [math]\mathbb{N}=\{0,1,2,3,...\}[/math] is given the symbol [math]\omega[/math], which denotes its order type. If you removed 42 from [math]\mathbb{N}[/math] and stuck it on the end (after the infinity of numbers), you'd have the same set of elements but it would look like [math]\{0,1,2,...,41,43,44,...,42\}[/math], and this is given the order type [math]\omega+1[/math]. You can separate out the odds and evens similarly and end up with [math]\omega+\omega=2\omega[/math]. This operations allow you to define standard arithmetic operations on infinities relating to orders, and similarly for cardinals.
In the first case, infinity is a shorthand for a limiting process (the infinity is hidden in the quantifier 'for all epsilon'), in the second case infinite objects are referred to explicitly.
Does that mean that in the second case "infinite" is being used as a quantity?
In the first case it's easier to think of as a direction. In the second case - for cardinals - they give the size of infinite sets, so yes they are probably quantities since they represent the magnitude of something.
Also, a x quantity of filler stuff fill in a x quantity of fillable things.
For example, two shoes fits in a box, if you increase the quantity of shoes to 3 shoes, it won't fit any more because it crosses the limit, it can also happen in the negative way. The thing is that with you have an infinitely large box, with an infinite amount of shoes in, no matter how many shoes you take off from the box, it won't change nothing. So talking about quantity doesn't make sense any more.
Is this a criticism of the epsilon-delta and epsilon-N convergence/continuity criteria?
"The wise man doubts often, and his views are changeable; the fool is constant in his opinions, and doubts nothing, because he knows everything, except his own ignorance" (Pharaoh Akhenaton).
Ok. Well epsilon-N implies 0.9 recurring = 1 anyway. AFAIK it's even true in non-standard analysis. 1 - infinitesimal isn't the same thing as 0.9 recurring.
East is not a location, destination, or even an obtainable goal. It is a direction relative to the current position and might more properly be stated as "east of here," wherever here may be, in the same sense that x + 1 is not an absolute quantity but instead something greater than x.
This is just... no. Look, even if I take your definition of quantity, I can easily show infinity is a quantity. Take the set of Natural Numbers (o, 1, 2, 3...). In set theory, the concept of "size" is formalized as what is known as "cardinality". The cardinality (size) of the set of Natural Numbers is infinity, specifically aleph-null. QED. You can say the Natural Numbers have "no limit" in the sense that it can always get bigger, but that doesn't mean it's impossible to quantify.
A better way to think about it is there are different kinds of infinite numbers, some larger or smaller than others. The set of Real numbers, for instance, is a larger infinity than the infinity of the Natural numbers. Cantor proved this with a proof by contradiction. No one is contradicting themselves saying there are infinite quantities.
But it's ambiguous (hence the [math]\sim[/math]).
As it turns out there's more than one infinite, there are infinitely many different infinites, no less (Cantor).
Anyway, Dedekind and Tarski came up with different (general) definitions that can be shown equivalent.
Very nice post.
A quibble. I just happen to be brushing up on ordinal arithmetic this week. Ordinal multiplication is defined backwards from our intuition in my opinion. [math]\alpha \times \beta[/math] is defined to be [math]\beta[/math] copies of [math]\alpha[/math] concatenated.
So for example [math]2 \omega[/math] means [math]\omega[/math] copies of 2 strung together. The ordinal 2 represents the order 0, 1. If you line up [math]\omega[/math] of those, you get ... drum roll ... [math]\omega[/math].
On the other hand, [math]\omega 2[/math] is two copies of [math]\omega[/math] side-by-side. You can visualize this as 0, 2, 4,6 , 8,...,1, 3, 5, 7, ... the evens-before-odds order. That's [math]\omega + \omega[/math].
Other than that quibble, great post.
https://en.wikipedia.org/wiki/Ordinal_arithmetic#Multiplication
Nothing to prevent you from adding 1 to infinity.
Seriously, you can even add infinity to infinity. Plenty of cases where that happens in mathematics.
Do you understand though?
Quoting Mr Phil O'Sophy
So, we have established that you DON'T understand it.
Quoting Mr Phil O'Sophy
Repeating an error ad infinitum does not correct it.
Quoting Mr Phil O'Sophy
And some of those are bigger, infinitely bigger, than the others.
Quoting Mr Phil O'Sophy
You have never studied mathematics.
Quoting Mr Phil O'Sophy
Indefinite in number, you say.
You have no rebuttal short of "I don't understand this".
Quoting Mr Phil O'Sophy
So it's you versus Cantor?
Quoting Mr Phil O'Sophy
Demonstrating your lack of comprehension does not constitute an argument.
Quoting Mr Phil O'Sophy
Primary school doesn't count.
Quoting Mr Phil O'Sophy
You don't have an argument.
By the way, I'm pretty bad at math, so don't take my word for it. I should just stop before I spread false information.
How was it unspoken if I literally said the assumption (the the natural numbers are infinite)? That aside, you aren't making sense. That the natural numbers do no, for instance, include the Real Numbers does not entail that the set of Natural Numbers is not infinity. In mathematics, infinity is not (as you claimed) defined as "having no limits". In this case that's especially obvious, because by "limit" you're already sneaking in the assumption of finitude (e.g. the natural numbers are finite, somehow, because the set doesn't include other types of numbers). This argument makes no sense.
You are simply ignoring the definition of infinity that mathematicians use and thereby conclude that it's incoherent because of we assumed your definition we'd get a contradiction. QED, your definition is wrong because it leads to a contradiction. That's ridiculous.
Your argument makes an obvious assumption, namely that all infinite sets are of the same size That's quite literally rejected in mathematics. Infinite sets which are countable, like the natural numbers, have the ability to be put into a one-to-one correspondence with a proper subset of themselves, e.g. we can map all the even numbers onto the set of natural numbers. Uncountably infinite sets (e.g. the reals) cannot do this mapping with the natural numbers, entailing that such sets are larger. Your definition leaves no real ability to use infinity in mathematically useful ways, e.g. Calculus.
Incorrect. The natural numbers are the counting numbers, so they do no include the reals. That does not entail the Natural Numbers have a finite *cardinality*, it simply means the set of natural numbers leaves out particular types of numbers. This simply means the set of natural numbers has a particular size of infinity.
The existence of an actual infinity (vs a potential infinity) is controversial among philosophers. I'm of the opinion an actual infinity cannot exist. I feel strongest about the impossibility of an infinite past, because that would entail a completed infinity: how could infinitely many days have passed?
Physicists accept the possibility of infinity in space and time simply because there is no known law of nature that rules it out. That doesn't imply the philosophical analysis is wrong, it just means that we don't know of any particular limits.
My opinions are consistent with the dominant opinion among philosophers prior to Cantor's set theory, but that doesn't seem like a very good reason to believe an actual infinity exists in the world.
Um, before every day there is another day. QED. Or to put it more directly, the cardinality of the set of days prior to day "n" can be put into a one-to-one correspondence with the members of the set of natural numbers. Ergo, the number of past days are infinite. I don't know if this is actually true, but there is no logical argument against the *possibility* of it.
However, this wasn't even really what I was suggesting. Between any two moments of time there's another moment. That's what I had in mind. And it's even clearer with the divisibility of space. It's nearly always taken to be a continuum, meaning it would be infinitely divisible.
What philosophical analysis? If we are adopting perfectly standard mathematics (or even most non-standard math systems) there is no contradiction whatsoever in supposing the past days are infinite. This will play into a bigger point I make at the end.
I hope it doesn't come across rude, but that just reads as "If you ignore the last 150 years of mathematics most philosophers would agree with me". Well that's... a defense anyone can make to defend their belief in whatever.
Look, my broader issue/point is this. The interplay between our beliefs about the world and the formal tools (maths, logics) is more complex than often made out (i.e. the influence goes both ways). However, generally the idea is that our physics needs math to guide it's conjectures, and our beliefs about the world ought to be in line with the dominant physical theories. If maths has explicated infinity as a coherent, precise concept - and it has - then presumably it becomes irrational to say (as I understand you to be saying) that "Yea yea, there's infinity in mathematics and in physics, but if you try to apply it to real things it entails a contradiction." I just don't get it.
Infinity is not a contradictory concept, so how is it supposed to produce a contradiction if applied to real things? Or is it supposed to be a category mistake? But how does that work? We talk about infinite collections in mathematics all the time, it's central to set theory. That doesn't mean infinite collections (or other infinite whatevers) can exist in our universe, just that you cannot rule them out as incoherent and thus fail to obtain in every possible universe.
I believe that beyond time and space, that infinity is only a mathematical construct.
We can not deal with infinite time or space, and nothing else in our world becomes infinite.
We measure the totality of energy by the rate of electrical flow times the amount of time, giving us Kilowatt hours.
If we multiply an infinite universe times the infinitely small, moving point in time in which it exists, perhaps the infinities cancel. This seems like a comfortable, perhaps pathological workaround to the notion that nothing can be infinite.
I would say that it simply isn't countable or computable. Yet it does in my view quite clearly define a quantity.
The problem is that math starts from counting. And hence we have all the problems with infinity.
Ammount of real numbers = infinite
Ammount of even real numbers = infinite
Does this mean that infinity divided by 2 equals itself?
Does this not violate the definition of a real number? Seems like there was something on this in "analytical geometry"...
I meant whole, or natural numbers, not real numbers.
Different levels infinities have different sizes. They're size is definite. "Infinity" is not one value.
The real numbers have a cardinality (size) larger than that of the even numbers. The even numbers have the same cardinality as the natural numbers, aleph-null, and so are "countably" infinite. The real numbers cannot be put into a one-to-one correspondence with the natural numbers, being that the reals are a larger infinity. Hence the reals are "uncountably" infinite.
Well, infinity is a very useful mathematical concept then. After all, the number "3" doesn't physically exist either.
3 things can exist but an infinite amount of things can not.
Space is definitely a Thing. There can be the 3D Space that we are familiar with, but there can also be 4D Space for example. 4D Space is a whole different Thing than 3D Space. If Space can be different Things then there can be no Space. That would be Absolute Nothingness.
It is not known if Space is Infinite or not. It depends on what the value of a particular constant in Physics is found to be. The current thinking is that our 3D Space is Finite but Unbounded in the sense that it curves back around on itself in some way.
You're other point wasn't what I was talking about. I'm not saying space is infinite in breadth, but it can be infinitely divided without hitting some kind of base unit or boundary point.
I wouldn't be so sure that Space can be Infinitely divided. I'll give that a Maybe. It might be subject to the Planck Constant. I don't think you should assume that a Physical Space is the same as a Mathematical Space.
- There is no number X such that X > all other numbers
- Because X+1 > X
Space is discrete that’s why we get paradoxes when we assume it’s continuous (Zeno’s paradoxes).
The Zeno paradox, where to walk any finite distance you have first walk half the distance, then half of the remaining distance etc. is a paradox because Zeno is throwing in a false assumption that you are going across each Half in the same amount of time. This is artificially slowing you down. Of course this would make it impossible to cover the total distance when you consider an Infinite amount of Halves. Zeno forgets that at the same time the Half distances are going to Infinity the number of Halves you are traversing (at constant velocity) is going to an Infinite amount of Halves per second. The Infinite Halves per second and the Infinite number of Halves are compensating Infinities that cancel the paradox.
Still I’d argue for discrete spacetime on the grounds:
- there is no such distance as 1/oo mathematically.
- Imagine a particle moving over a finite period of time. Continuous spacetime would require the particle to have occupied a actually infinite number of states which is nonsensical.
Still even if space is continuous, that would only be a potential infinity rather than actual infinity.
I say that Infinity is a Mathematical Fiction that only exists in the world of Mathematics. But since the construction of the Zeno Paradox uses Infinity as the basis of the argument we must accept the premise and argue from that. I think the compensating Infinity argument is the best way out of the Paradox.
If a particle only occupied discrete states then according to your theory it would have to jump from position to position while moving. It would necessarily have to stop at each position for the time it would take to continuously travel between two of the positions. This is as nonsensical as a continuous movement with Infinite intermediate positions. These are both nonsensical and serve to illustrate the problems you can get into when you think a little more Deeply about things. Good Thoughts however.
A good point. It depends on your view of time as to whether you think the particle exists in an actually infinite number of states:
- Presentist. The past does not exist. So the particle does not exist in an Actually Infinite number of states, just one state, the present.
- Eternalist. The past exists so continuous time implies the particle must exist in an actually infinite number of states.
Presentism leads to paradoxes, so that suggests Eternalism. But time must be discrete for Eternalism to be free of Actual Infinity (which I class a paradox).
The Infinitely Large and the Infinitely Small are a real pain in the Brain.
Now imagine a Square that is the smallest Square that is not equal to Zero. This thought sends your mind into an endless recursive loop of the Square getting smaller and smaller and we soon realize that it is impossible to imagine such a smallest Square. One thing we can say is that this Square is Infinitely small but is still a Square. In general mathematics this would be called a differential Square or an infinitesimal Square.
Next imagine the Square that was drawn on the paper growing larger and larger. If the Square was exactly in the center of the paper the sides of the Square would eventually move off of the paper and past the edges of the universe. It remains a Square no matter how large it grows. If we stop growing it and start shrinking it back we can bring the Square back to the original size. But now imagine the Square growing to Infinite size. The sides would all move out to infinity. No matter how far you went in the universe you would never encounter a side of the Square. The Square has effectively exited the universe. We could also grow a Triangle in the same way and at Infinite size it will no longer be found in the universe. The Square and the Triangle lose their identity when they are Infinite size. They become something different. Paradoxically they become something less than what they were. You might think that the Square and Triangle are still out there at Infinity. But there is no "there" at Infinity. The Square and Triangle are gone. If you think you can go out "there" to an edge of the Square or Triangle at Infinity then that "there" is not Infinity. Infinite size is an unrecoverable threshold of size that changes everything.
Now imagine a Square that is the largest Square that is not equal to Infinity. Similar to the differential Square, this thought sends your mind into an endless recursive loop of the Square getting larger and larger and we again soon realize that it is impossible to imagine such a largest Square. We can say that this Square is Infinitely large but is still a Square that exists in the universe.
I think that just as Infinite Squares are not possible it is probably true that any Infinite Physical quantity of anything is not possible. Just because an equation in Science goes to Infinity, it doesn't mean that the Physical quantity in the equation is able go to Infinity. I think this is a limitation of what we can do with Mathematics. Seems like a minor limitation but it has big consequences when equations in Science go to Infinity.
A good point. Both the process of growing something to oo and shrinking something to 1/oo are destroying information, which is not meant to happen in the physical world.
Hence infinity, through the line as one considering one premised in empirical qualities observes all numbers as an observation of linear qualities through time as directive in nature.
The line as 1 is infinity as 1 with one itself being infinite in the respect it is continuous as existing through itself as itself. We can observe that a line as 1 through 0d space or void must effectively projects if it is to exist. However considering the line has nowhere to project but must project to something if it is to maintain its nature, the line must effectively fold through itself by multiplying in directions.
In these respects the 1 original line as infinite must individuals into multiple lines which each line being a ratio relative to each other and the original line while being infinite and one in itself. Finiteness is strictly the relation of infinities with infinity, as an absence of finiteness, being an absence of relation with infinity and finiteness existing at the same time in different respects due to there positive and negative qualities.
Not only is infinity quantifiable but there is one infinity and multiple infinities with these multiple infinities being the premise for finiteness or time.
In simpler terms the 1 directional nature of the infinite line shows infinity it only as quantifiable but effectively existing as a limit.
As far as I know, infinity is a quantity, so a number, that has no fixed value. It goes on forever.
Think of * as a number. An infinity of * is still a collection of * but we can't fix an exact value to it.
Basic math operations (+ - × ÷) fail with infinity.
I guess we aren't supposed to use it that way.
Is a bishop a religious guy with a cool hat? Or is a bishop a piece in a certain game that moves diagonally? There are connections between the hat-guy and the chess-piece, but they are different, and they both make pretty good sense in their context.
I think it's the same with infinity.
Quoting fdrake
Since you said correctly that in the definition of limit the notion of infinity is hidden in the quantifier I think you are not confusing the limit to infinity (infinity indicates the graphical correspondent to the behavior of a function, in so far as its values become arbitrarily large) with infinity of the limit(infinity limit a process) I suggest to not use the misleading terms: 'limiting process' but 'unlimited variables bounded in regards to which the limit is considered'.
Also Quoting tom
Of course tom is right(but unfortunately unsuccessful) in preventing the tedious error of conceiving of infinity as a set of numbers. More explicitly: infinity is a relational concept, and its use as a factor just means the operations operate on variables(and infinite variables[of numbers] plus the unity is of course a not problematic thing). One thing is infinity as relational concept, another is the Symbol of infinity to indicate infinitely many variables.
Quoting fdrake
I don't think cardinality offer a quantitative view of Infinity, since it is either a relation between a set and its elements or between its elements and numbers(e.g. a set is D-
infinite iff for every natural number the set has a subset whose cardinality is that natural number) or between sets(e.g. the cardinal of R is bigger than the cardinal of I)
Numbers are sets, in the usual axiomatizations, so cardinality very naturally fits our idea of quantity.
The only problem here is that "sets" are based in qualities, and there is a conceptual difference between quality and quantity. Therefore set theory does not naturally fit our idea of quantity. So set theory provides a set of axioms which modify mathematics in a way so as to be inconsistent with our natural idea of quantity.
This is false. There are sets of numbers, but number themselves are not at all sets, just as there are cluster of berries, but no berry is a cluster.
This is false. Sets are based on RELATIONS(between something, i.e. a set, and its elements. The empty sets have a relation such that no elements belongs to it).
Quoting Metaphysician Undercover
These are YOURS(wrong) assertions or opinions about what intuitive quantity should be and how this relates to set theory and about set theory.
Exactly. Nonetheless this DOES NOT MEAN NOR IMPLY infinity IS a quantity. Is a relation.
I will open a thread about it, because it is UNBELIEVABLE that people still think(like Medieval theologists and the dogmatics) infinity as a quantity.
A relation is a comparison between a quality of one, and a quality of another. Therefore it is inherently qualitative.
Define explicitly a quality without presuming a quantity man, if you can. You will find you can not, and since quantity presupposes a relation, insofar as a quantity is relation between spaces, then quality presupposes relation.
I add: YOU think relations are comparison, but this is a very special case of relations. I'll give you a fair example of a relation which is not a comparison: a function. You set your prejudices as metaphysical truth, and this just because your ignorance of the advancements or acknowledgements of mathematics and logic.
Quality is the degree of excellence of a thing. This implies a relation, and as I said, relations are inherently qualitative. However, it does not imply a quantity, as quantity requires discrete units, and "degree" may be applied to a continuum. It is only when we refer to individual "degrees" that quantity is implied in such a relation.
Of course they are.
E.g. 1=0?{0}={0}={{ }}
This is the 'singoletto'(I don't know the English word, kind of 'single set') not of a number.
Quoting MindForged
The cardinality and ordinality are not at all variables which can be 'put into a function'. You are very confused.
Quoting MindForged
This is not even wrong. It has no sense. Numbers are not defined otherwise than by postulating a number and stating which operations preserves that being a number. See basic, high school Math(Peano Axiomatization of Arithmetics).
Quoting SophistiCat
There is no reply to a deep confusion like yours. BELIEVING numbers be sets is at least a funny thing.
Quoting Metaphysician Undercover
Frankly, a definition so senseless as all the medieval definitions were. You are deeply confused. You can quantify on something, by selecting somehow a unit of quantity, only if there is yet a quantity independently from that selection.
Furthermore it does not give us the 'capacity of measuring', because, in fact, a measure(as you may intend, for the use of terms by you is ambiguous: physical measure? or which one?) presupposes a mode of measuring, and so presupposes a quantity on which that method is to be applied. In physics, this is matter, or whatever you may call it.
But since you are so darkened on thoughts about what a quality is I will give you a fair dispute, stating better your naive argument and than responding to this peter version(be honored: the argument is by Kant):
«Reality in appearances(i.e. matter as we perceive it) always has a magnitude(a scale of degrees, i.e. quality), which is not, however, encountered in apprehension(you do not apprehend the space in which it is by itself) , as this takes place by means of the mere sensation in an instant and not through successive synthesis of many sensations, and thus does not proceed from the parts to the whole; it therefore has a magnitude, but not an extensive one.» p.290 of Critique of pure reason (guyer wood)
Then he continue:
«Now I call that magnitude which can only be apprehended as a unity, and in which multiplicity can only be represented through approximation to negation = 0, intensive magnitude. Thus every reality in the appearance has intensive magnitude, i.e., a degree. » ibid.
Now, if something is to be selected as a discrete unity, as you say, it presupposes a quantity on which this selection is operated. Or do you think we actually create quantity by itself, against the basic postulate of physics?
Kant is saying that we do not apprehend it, in the sense that the homogeneity of the synthesis is a presupposition, and also it renders possible at all to select a unity: for a unity is a unity of a manifold, and if there were no homogeneity it would not be the unity of that manifold.
It is only deniable by pathologically affected(in the brain) people that homogeneity is a spatial property. And quantity in general if we have to be cautious, thus not saying quantity in general is space(which is a big assumption), we say that PRESUPPOSES ONLY SPACE, as its parts are external to one another: this sufficing to provide an account of quantity, on the postulate(common to every physical theory that has any sense) that space is occupied by something(I,e. matter), and THUS a quantity is possible at all, insofar as we can distinguish it from the space itself and thus it is possible to be measured with a certain referential unity.
Hence as the scale of degrees(quality) relies on a spatial property, than the degrees do rely on that to. But that property alone can not give us any quantity, for, by a quantity, we do not intend a mere externality between parts, but an OCCUPATION of (parts of) space. Hence Space(spatial properties we are able to detect) and matter are presupposed by any concept of quality that has any sense.
But neither Space nor matter presupposes any detectable quality by themselves. Quality, furthermore, presupposes a RELATION between space and something, which renders possible to detect some spatial properties or, as you seem to prefer, to select from that properties units, in respect to which establish a scale of measuring. This something is matter. Matter does not imply quality(degrees) but the distinguishability of degrees implies matter. But matter presupposes space. Then quality presupposes space. Either you identify space with the properties we can distinguish and classify under the kind 'spatial' and name IT quantity, or you do not identify space with those properties and call those properties 'QUANTITY' it is the same for our question: quantity it is presupposed by quality.
The word you're thinking of is "singleton" I believe. And no, that's not just the singleton of a set. It was the definition of the number one in set theory. If Zero is defined as the empty set, One can be defined as it's successor; the union of Zero with singleton-Zero.
I'm still confused about what you meant when you said I had a medieval theologian type of thinking...
This is the condition to CONSIDER the 'singleton'(I trust you on the term) as the number one, but not to DEFINE it. Numbers are defined by a postulate and succession: see Peano Arithmetics( 0 is number(POSTULATE). If n is a number, then n+1 is number.(AXIOM) Then 1 is a number, as 0+1=1.)
Quoting MindForged
Not referring to you in particular. By 'medieval thinking of infinity'(and conception more or less related to it) I mean a very precise thing that I say to you.
Thinking infinity has THE QUANTITY OF WHICH NO QUANTITY IS GREATER is the MEDIEVAL conception of infinity.
Since Kant gave (1781!!) an explanation on why this is wrong, I cite from him(Guyer Wood translation, p.472)
«I could also have given a plausibleb proof of the thesis by presuppos ing a defective concept of the infinity of a given magnitude, according to the custom of the dogmatists. A magnitude is infinite if none greater than it (i.e., greater than the multiplec of a given unit contained in it) is possible.58 Now no multiplicity is the greatest, because one or more units can always be added to it. Therefore an infinite given magnitude, and hence also an infinite world (regarding either the past series or ex tension), is impossible; thus the world is bounded in both respects. I could have carried out my proof in this way: only this concept does not agree with what is usually understood by an infinite whole. It does not represent how great it is, hence this concept is not the concept of a maximum; rather, it thinks only of the relation to an arbitrarily as sumed unit, in respect of which it is greater than any number. According as the unit is assumed to be greater or smaller, this infinity would be greater or smaller; yet infinity, since it consists merely in the relation to this given unit, would always remain the same, even though in this way the absolute magnitude of the whole would obviously not be cognized at all, which is not here at issue.»
I will, a time or another, explain in details how mathematical infinity relates to what Kant meant by Transcendental CONCEPT of infinity, i.e. the relation between math-logical and epistemological infinity.
But my comment about numbers being sets (everything is a set in the set theory construction of mathematics - obviously) was made in the context of the preceding discussion, which Ikolos does not or will not follow.
How is this different from what I said?
Quoting Ikolos
Well I that's definitely not how I described infinity, and it doesn't even seem correct because it's too vague. As there are many sizes of infinity, I don't even think I could pretend to accept "the quantity of which no quantity is greater" as a description of infinity.
Hm, okay but I don't really get what their point about infinity was supposed to be. It didn't really sound like something which would accurately characterize how the concept is used and understood in modern math. Aleph-null is definitely not a quantity above all others, for instance.
It's the #1 definition in my OED under "quality". I really think that it's you who is confused. would you prefer:"a distinctive attribute or faculty: a characteristic trait"?
Quoting Ikolos
You clearly have this backward. Unity does not presuppose quantity. Quantity follows from unity, that's how we measure, by units. Quantity requires the individuation of units, therefore it requires unity. Unity does not presuppose quantity. So it s quite clear that we do create quantity, as it is minds that individuate units such that something may be counted. Which basic postulate of physics states otherwise?
Quoting Ikolos
Yes, I agree that "degrees" rely on "quantity". And a difference of quality is measured as a difference of degree, that's the point I made. This is what allows us to measure different qualities. When we notice difference, and we produce "degrees" to measure that difference, we produce the means to quantify that quality. The units of measure, "degrees", by which the qualitative difference is divided, may be a completely arbitrary production of the mind. And, this arbitrary division of the qualitative difference, into fundamental units, "degrees", may give us the capacity to measure the quality.
Quoting Ikolos
You are misrepresenting the difference which exists within the thing, matter, and replacing it with our measurement of that difference, which is with degrees. Degrees of difference are not necessarily distinguishable because "the degree" may be arbitrary. The premise of "the distinguishability of degrees" misleads you. So we commonly divide space by degree with no implication of matter.
This is naive set theory darling. Thank you for your opinions directly expressed on my communication skills. I am sorry you don't understand set theory, but that's not my fault.
Quoting Metaphysician Undercover
Yes, and what do you measure my friend? Quantity. Hence there is quantity yet, and you measure it. That fact that the methods and units of measurement may be relative(as you correctly say) does not make this less true, i.e. quantity is presupposed REALITER as what is to be measured.
Quoting Metaphysician Undercover
Independently existing matter, independently from particular modes of perceiving it, and that actually causes any perception to happen.
Quoting Metaphysician Undercover
The degree as a reference of measure is arbitrary, but clearly the degree of complexity of bacteria is inferior to that of an dog as an organism. No one divide space by degree man, that is nonsensical. Topology does not consider differences of space by degrees at all.
The differences in the things, as you say, is a difference which is not merely spatial, but involve matter, hence degrees. Because if you can not recognize the difference(with a possible measure) things are not at all distinguishable, except by logical or topological characteristics. Unfortunately, we are sensible beings.
Thank you for being rational. As you can now see, thinking infinitity as a quantity whatsoever is against any basic calculus teaching.
Quoting MindForged
Is different in a very significant way. Because CONSIDER something as something else(with reasons to do that) is a thing, DEFINE(as you said) something is another. Citing you:
Quoting MindForged
Numbers are not defined in set theory. They are considered elements of sets. Numbers are defined by induction in Peano Arithmetics, postulating 0 as a number and posing the axiom of succession.
No, this is clearly not the case. The units of measurement, which are counted as a quantity, are very often created by the human mind, for the purpose of measurement, they are artificial. So "quantity", as the thing measured, is not necessarily presupposed. In the cases where we cannot find individual units to count, we simply create them, giving us the capacity to measure without there being any real quantity which is being measured.
We're talking about measuring quality, which is said to differ by degree, and the problem is that the units of degree are often artificial and may be arbitrary. For instance there are degrees of temperature. The thing measured is a difference, the unit of measurement, one degree, is artificial and arbitrary. So the quantity is artificial, and there is no actual quantity which is being measured, the "quantity", as the measurement, is assigned to it, arbitrarily, it is not something within the thing measured.
You can also look at the 360 degrees of a circle in the same way. That number of degrees around a circle is complete arbitrary. The circle could be divided into an infinite number of degrees, and there could be an infinite number of degrees between each of the four right angles, and any other angle around the circle, expressing a continuity around the circle, rather than the discrete units of degree which are commonly used. This is actually expressed with minutes and seconds. There is no real quantity of measurable units which are being measured around the circle.
Quoting Ikolos
That's a postulate of physics?
Quoting Ikolos
Have you not heard of a circle?
1. Infinity is a never ending quantity.
2. Infinity is not a number.
Infinity is a symbol given to an unlimited amount of things.
Btw, countable and uncountable infinity is counterintuitive, insane and nonsensical. Cantor was a lunatic.
In the diagonal proof you have an infinite amount of rows and columns. Supposedly you can make a new infinite number different from the one on the list. How? By taking the first digit and changing it to whatever, then the second digit, and so on. This is nonsense.
First of all the new number has conditioned itself to be infinitely different from the one on the list. In no possible way can the number be the one on the list, however the same can be said by the number on the list ever actually being a quantity. It has and will never end. Thus they're both infinite numbers, and thus they both have the same length. The only difference is the rate at which they grow.
Second of all the rate of growth in infinite numbers is not a valid argument to define N (aleph-null) as an ordinal 'uncountable infinite'. This is because the rate of growth can still be achieved without ever actually having the list to being with. This new number N is in no way different or special from the numbers on the list.
Third of all the rate of growth to infinity does not change at all the fact that they're both infinite. Saying that some N1 is infinite will then have an N2 being infinite proceed from that one makes no sense and is mathematically possible but absolutely irrational, and a waste of time for your brain to acknowledge. But people don't understand these three simple concepts.
Maybe it signals an inability to quantify.
Inability to quantify or a never ending quantity I guess could be considered the same.
Planck Length, Planck Scale, speed of light (which is basically a scale constant) are not in any sense arbitrary. Unless Im mistaken, the SI system is based on non-arbitrary physical constants not this nonsensical notion of "qualities" or whatever. And even if it isn't, such a thing is possible but the gain is little for all the work required. See Natural units.
Quoting Metaphysician Undercover
That something could be done a different way does not make something arbitrary. There are perfectly sensible reasons to put the number of degrees at 360. It's a highly composite number allowing us to avoid fractions (which are hard for humans to do, hence the preference for decimal expressions), it's not a large whole number so it's fairly easy to do basic math with (particularly division), etc. I'm thinking you're using a weird definition of "arbitrary" or not explaining why it is (supposedly) so.
#1 is false in some sense. Unless you ignore modern math, there are a hierarchy of infinities. #2 is extremely wrong, there are many infinite numbers. The transfinite cardinals and ordinals will fit any sensible, non-arbitrary definition of a number.
Quoting Emmanuele
Except the Diagonal argument is a proof by contradiction, the most standard argument type of all.
Quoting Emmanuele
Who cares if it's "conditioned"? That is not a criticism, you're whining with this non-objection. The mere fact (proven by the opposite entailing a contradiction) that such a number necessarily exists is why mathematicians accept Cantor's diagonal argument. "Never actually being a quantity" is, naturally, something you do not demonstrate to be the case. That the number "never ends" does not mean they have the same length, that's a non sequitur. Show exactly where the "conditioned" number will appear in the original set. That you cannot (again, because it would be a contradiction) entails the different sizes of the respective sets.
Quoting Emmanuele
Aleph-null is not an ordinal, it's a transfinite cardinal. Literally the smallest, first transfinite cardinal is aleph-null. And thus it is not uncountably infinite, it's countably infinite, because the definition of that is "being capable of being put into a function with the set of natural numbers".
Quoting Emmanuele
"Both infinite" says nothing more than that they're both the same type of number. That "0.33" and "0.44" are both real numbers doesn't mean they're both the same number. You are under a terrible misapprehension. If the sets were the same size they could be mapped onto each other. That's how size is defined in math, they have to have the same cardinality. But of course, if you cannot even accept Cantor's proof by contradiction that the naturals and the reals cannot be the same size then I'm not surprised you fail to grasp this.
Cut the nonsense. Show exactly where the sets map together to make them the same size. That you cannot means they aren't the same size. Ergo, one is larger and the other smaller. It's funny you say it's mathematically possible yet "absolutely irrational". I'm sure you can defend that characterization as coherent. One would assume mathematics cannot be incoherent, given incoherency in mathematics means triviality; everything becomes a theorem if the math is incoherent, but there's absolutely no evidence that current mathematics is trivial. Show the formal contradiction. You'll win a Fields Medal.
As I said, and provided examples for, some divisions of degree are arbitrary, I didn't say that all are arbitrary. But the fact that some are, is all that's required to disprove Ikolos' claim that quantity is what is measured. Actually, quantity is the measurement.
Quoting MindForged
That's not even an argument, the number of degrees in a circle is not arbitrary, it was chosen because it's "easy to do basic math with". The principle divisions of the circle are the four right angles. So the number of degrees in a circle need only be divisible by 4 in order that "it's fairly easy to do basic math with". That the circle consists of 360 degrees, and not 4, 8, 12, 16, 20, 400, 800, or any other number of degrees divisible by 4 is arbitrary. That the number of degrees in the right angle is 90, and not 1, 2, 3, 4, 5, 100, 200, or any other number, is completely arbitrary.
Incorrect, their argument was that some were not "qualities" as you deemed them because they are part of reality. Pointing out that some aren't (as that user already admitted) is very much besides the point when they already admitted so.
Quoting Metaphysician Undercover
How is that not an argument? Ease of use is a perfectly legitimate reason to do prefer something. You user particular kinds of screw heads in because they're easier to use for particular kind of screws (this generalizes to tools in general). Hell, the entire reason a radian is defined as 2 x Pi is because it makes calculus and trigonometry easier if angles are measured in radians instead. Again, you're working under a strange definition of "arbitrary" if practicality doesn't count because it's very helpful that a circle has 360 degrees. Also, try to do set a circle equal to 4 degrees and see how the math works out for you. Other numbers have many divisors, but they aren't as useful mathematically (example, 2570 can be cleanly divided by every number 1 - 10 but it's not very useful in geometrical calculations to set it that way).
That's a very good damn response.
Logically I would state that the reason why aleph-null makes no sense is because like both aleph-null and the numbers on the list are infinity, then to say that aleph-null is in any way different from the numbers on the list is to state that the numbers on the list have an ending. This is because for them to actually be different is for both numbers to end in a difference. Such statement may sound illogical at first glance because no matter what number appears it will always be different, however the point is that if we think about it aleph-null is still impossible to fully become different.
All the notation that comes after aleph-null is to assume the difference. Here is a mathematical intepretation of my reasoning.
j = {0, 1, 2, 3, 4 ... M}
N = {1, 2, 3, 4, 5 ... X}
j = numbers on the list
N = Aleph-null
X= Statement of difference
M = hyperreal of j
however
N = {1, 2, 3, 4, 5 ... X },
N = {1, 2, 3, 4, 5 ... M1 ... X ... },
N = {1, 2, 3, 4, 5 ... M1 ... M2 ... X ... },
To say that aleph N is different from j is to say that j ends at some M for it to become X. It doesn't and never will become X (different).
Even mathematically, to state that aleph-null is a thing is to reduce the nature of the problem to cardinality. Which is meaningless. The infinity to be reasoned in any physical sense is to acknowledge that impossibility of convergence to some difference.
Also, the reals are bigger than natural numbers because they're defined in terms of themselves. They are not conditioned to exist to something which will never end. In this case aleph-null is conditioned to exist for when M ever becomes X - It will never do such a thing.
That's a very good response.
Logically I would state that the reason why aleph-null makes no sense is because like both aleph-null and the numbers on the list are infinity, then to say that aleph-null is in any way different from the numbers on the list is to state that the numbers on the list have an ending. This is because for them to actually be different is for both numbers to end in a difference. Such statement may sound illogical at first glance because no matter what number appears it will always be different, however the point is that if we think about it aleph-null is still impossible to fully become different.
All the notation that comes after aleph-null is to assume the difference. Here is a mathematical intepretation of my reasoning.
j = {0, 1, 2, 3, 4 ... M }
N = {1, 2, 3, 4, 5 ... X }
j = numbers on the list
M = hyperreal of j
N = Aleph-null
X= Statement of difference
however
N = {1, 2, 3, 4, 5 ... X },
N = {1, 2, 3, 4, 5 ... M ... X2... },
N = {1, 2, 3, 4, 5 ... M ... M2 ... X3 ...},
To say that aleph is different from j is to say that j ends at some M for it to become X. It will never do such a thing.
Now, the real numbers are greater than the naturals because they're both defined by themselves to be so. In this case aleph is defined to exist as long as it satisfies the difference it has with j. It doesn't have one because it cannot happen due to the nature of what is infinity.
Mathematically you can reduce all this into cardinality but it throws out the window the sense of infinity in any physical and reasonable sense, and so it defines things in a finite way similar to a hyperreal. Stating that it exist if one day M stops growing to become X.
Even then the argument behind a hyperreal is more logical then that of aleph-null. Because the hyperreal is taken into account to never be n. And so, to be able to stay at some n.
The difference X involves taking some n of j to be x. Which is to state that M is an n. No logos my buddies, what the heck is going on?
With this I say that these mathematics are physically inconsistent. So you'll never see anything like this in the physical world. To think this is amazing is to be amazed at 2 + 2 = 3 but in a more complex and sophisticated way
That does not follow. For one to be larger than the other all that need be true is that one set has a greater cardinality. What this will mean is that when you try to place them in a one-to-one correspondence with each other, it fails to be possible to do so. After all, sets that can be mapped together in this way are the same size. What Cantor showed was that it's impossible to map the naturals with the reals on pain of contradiction, it turned out the reals were larger not that the naturals had an end (in the sense of a final member). That's what makes them different, despite being infinite. They're different levels of infinity.
I get the point. The ending member is different in a one to one correspondance from T (numbers on the list) to N (aleph). I would just like to believe that from R to N (naturals) the last member is different because that's how they are defined by themselves. If we make a number out of the difference of a specific number is to state that the former was a number at all, even in regards to the cardinality of a set.
I would like to believe that this up here is sufficient to say that aleph as a set was to just be math dribble. And the last member of the set T never existed in order for it to be different. So the definition of the set aleph is not satisfied.
I can see that one, or both of us, misunderstands what Ikolos was saying. I can restate what I was saying though. I said that qualities can be quantified, but it is a mistake to attempt to qualify quantities. Both qualities and quantities are "part of reality", so this reference is just a diversion. The issue is what the mind is doing when it attempts to quantify a quality, or qualify a quantity.
Ikolos replaced "quality" with "relation", and I had to insist that relation is a quality rather than a quantity, because Ikolos wanted to argue that a relation is a quantity, without the required mental act which quantifies that quality.
Quoting MindForged
As I said, there are many other numbers which offer equal, or greater ease of use, so there is no basis for your argument.
Quoting MindForged
If a circle were 4 degrees, I see that an acute angle would be less than one degree, and an obtuse angle would be greater than one degree. Forty five would be half a degree, and one eighty would be two degrees. Looks very easy to me. I'm no mathematician so I might be missing something. Where would the mathematical problem be, which would make 360 degrees mathematically easier than 4 degrees?