Infinite Bananas
We have two rows of identical bananas and each row stretches out to actual infinity. The two rows are lined up so that there is a one-to-one correspondence between them. So the two collections of bananas are therefore identical:
{b, b, b, b, …}
{b, b, b, b, …}
We add one banana at the start of the second collection and then shift all the bananas right one place so they are lined up again - they are in one-to-one correspondence so they are still identical collections.
We then remove every second banana from the second collection and then shift all the bananas in the second collection to the left so they are lined up with the first collection again. Again they are in a one-to-one correspondence so they are identical collections.
What we have just done with bananas is of course:
?+1=?
?/2=?
So if you believe in actual infinity, there exists objects that obey the following axiom:
When it is changed, it is not changed
Surely this statement is a contradiction? Surely in our reality, when something is changed, it changes?
So I think we have to conclude that actual infinity is not part of our reality (or indeed any logical form of reality), it is just an illogical concept that exists in our minds (along with concepts like levitation, talking trees and square circles).
--------------------
(above is my revised argument. Below is my original post... which has errors...)
We have two rows of identical bananas and each row stretches out to actual infinity. The two rows are lined up so that there is a one-to-one correspondence between them. According to Cantor, the two sets of bananas are therefore identical sets.
We add one banana at the start of the second set and then shift all the bananas right one place so they are lined up again - they are in one-to-one correspondence again - so Cantor would claim they are identical sets.
We then remove every second banana from the second set and then shift all the bananas in the second set to the left so they are lined up with the first set again. Again they are in a one-to-one correspondence - so Cantor would claim they are still identical sets.
What we have just done with bananas is, of course, the transfinite arithmetic:
?+1=?
?/2=?
So in Cantor’s dream world (the alternative/virtual reality of infinite set theory), there exists objects that obey the following axiom:
'When it is changed, it is not changed'
Surely this statement is a contradiction? Surely in our reality, when something is changed, it changes?
So I think we have to conclude that actual infinity cannot be part of our reality (or indeed any logical form of reality), it is just an illogical concept that exists in our minds (along with concepts like levitation, talking trees and square circles).
{b, b, b, b, …}
{b, b, b, b, …}
We add one banana at the start of the second collection and then shift all the bananas right one place so they are lined up again - they are in one-to-one correspondence so they are still identical collections.
We then remove every second banana from the second collection and then shift all the bananas in the second collection to the left so they are lined up with the first collection again. Again they are in a one-to-one correspondence so they are identical collections.
What we have just done with bananas is of course:
?+1=?
?/2=?
So if you believe in actual infinity, there exists objects that obey the following axiom:
When it is changed, it is not changed
Surely this statement is a contradiction? Surely in our reality, when something is changed, it changes?
So I think we have to conclude that actual infinity is not part of our reality (or indeed any logical form of reality), it is just an illogical concept that exists in our minds (along with concepts like levitation, talking trees and square circles).
--------------------
(above is my revised argument. Below is my original post... which has errors...)
We have two rows of identical bananas and each row stretches out to actual infinity. The two rows are lined up so that there is a one-to-one correspondence between them. According to Cantor, the two sets of bananas are therefore identical sets.
We add one banana at the start of the second set and then shift all the bananas right one place so they are lined up again - they are in one-to-one correspondence again - so Cantor would claim they are identical sets.
We then remove every second banana from the second set and then shift all the bananas in the second set to the left so they are lined up with the first set again. Again they are in a one-to-one correspondence - so Cantor would claim they are still identical sets.
What we have just done with bananas is, of course, the transfinite arithmetic:
?+1=?
?/2=?
So in Cantor’s dream world (the alternative/virtual reality of infinite set theory), there exists objects that obey the following axiom:
'When it is changed, it is not changed'
Surely this statement is a contradiction? Surely in our reality, when something is changed, it changes?
So I think we have to conclude that actual infinity cannot be part of our reality (or indeed any logical form of reality), it is just an illogical concept that exists in our minds (along with concepts like levitation, talking trees and square circles).
Comments (204)
{1,3,5,...}
And the odd numbers without 1
{3,5,7,...}
are not the same set, the second is a proper subset of the first, but they're the same cardinality (the bijection is f(x) = x+2 where x is an odd number).
1. The first pair of bananas is in one-to-one correspondence
2. If the nth pair is in one-to-one correspondence so is the nth+1 pair
3. So all bananas are in one-to-one correspondence
4. So the sets are identical
instantaneous existential locations with respect to an observer....whence the ensuing word salad (or fruit salad :wink: )
I love this. Accusing math of being a Democrat.
Not quite.
The two rows are different in location, but the bananas are identical to each other in every other aspect but location.
Try the experiment now.
A set is a set is a well-defined collection of distinct objects, so my identical bananas cannot be said to form a set.
I still maintain however that my argument highlights the absurdity of actual infinity.
This is another one of those philosophical "problems" that is a misuse of terms.
How does one add a banana at the beginning of a row of infinite bananas? There is no beginning, and therefore no second banana, in a infinite row of bananas. There is no beginning or end with infinity. You're simply misusing terms.
What I mean is:
{ b, b, b, b, ... }
^
new banana is inserted here (at the start).
So the infinite collection has a start but no end.
An infinite collection has no start and no end.
So I think we are in agreement?
I believe time, space, matter/energy are all finite and discrete.
The first one:
Quoting Devans99
Firstly, if all the elements of the sets are identical, then they just have one element. Sets are defined by what distinct elements belong to them; a set is a collection of distinct objects. If x is in X, there's only one copy of it. If you want to consider set like objects that allow multiple copies of identical elements in them, that's a multiset.
[hide]Don't get hung up about sequences like {1,1,1,1}, these are formally distinct objects; they're sets of ordered pairs; {1,1},{1,2},{1,3},{1,4}. Sequences are functions from sets of natural numbers to other objects, you can write it out like a set because the reading order from left to right represents the sequence order neatly.[/hide]
The second one:
Quoting Devans99
Two sets being in a one to one correspondence says nothing about whether they are identical sets. The odds are in a one to one correspondence with the evens, but even numbers are necessarily not odd. However, if you stipulate the definition:
(C) Two sets [math]X,Y[/math] are related in sense C when and only when there is a bijection between them. When a bijection exists between the two sets, write [math]X \cong Y[/math].
You'll see that (C) is an equivalence relation. The equivalence classes of (C) are sets of the same cardinality. IE, insofar as C is concerned:
[math]\{1,2\} \cong \{3,4\} \cong \{5,6\}[/math]
But
[math]\{1,2\} \neq \{3,4\} \neq \{5,6\}[/math]
!
You'll also see that if two sets are identical, a bijection exists between them (the identity function), so a set has the same cardinality as itself. That is [math]X = Y \implies X \cong Y[/math], but the reverse does not hold. The reverse implication is precisely what you require to go from 3 to 4 in your argument, and you're obtaining it by equivocating between two sets being identical and a bijection existing between two sets.
You should have really read the posts above. I did acknowledge my mistake here:
https://thephilosophyforum.com/discussion/comment/367343
And I've also updated the argument in the OP accordingly. Sorry.
Quoting fdrake
Two collections of identical objects in a one-to-one correspondence are, by mathematical induction, identical collections.
For example, some folks believe that past time is actually infinite; implying an actually infinite collection of moments in the past.
Or some hold that space is continuous, implying an actually infinite collection of distinct spacial positions in a unit of space.
Assume what you're saying is true. Then {1} = {2}, then 1=2, contradiction, then what you're saying is false.
For example, many cosmologists hold that time has no start, implying an actual infinity of past time. All those past moments actually happened, so it would count as an instance of actual infinity, regardless whether you hold a presentist or eternalist viewpoint.
Then there is the axiom of infinity. It states that there exists a set with an actually infinite number of members (the natural numbers). As you have pointed out, no such set exists (outside our minds where the impossible is possible).
Quoting tim wood
I feel a line that has existence outside our minds must be constituted of something - points or sub-segments or such. The most common definition is that a line is a set of actually infinite points.
Quoting tim wood
Do you consider Cantor sane? If so:
"Accordingly I distinguish an eternal uncreated infinity or absolutum which is due to God and his attributes, and a created infinity or transfinitum, which has to be used wherever in the created nature an actual infinity has to be noticed, for example, with respect to, according to my firm conviction, the actually infinite number of created individuals, in the universe as well as on our earth and, most probably, even in every arbitrarily small extended piece of space. - Georg Cantor
Wiki: Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I.[33] The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had a fatal heart attack on January 6, 1918, in the sanatorium where he had spent the last year of his life.
Profound set theory can be harmful to one's health, Devons99!
Inaccurate notation, since by the axiom of extensionality, the set {b,b,b,b,b,...} is the exact same set as {b}. Perhaps if you notate it [math]\{b_1, b_2, b_3, \dots \}[/math] your argument will be more clear.
I don't think that helps any. You're starting out with bad notation and that's leading to incorrect conclusions. By collection do you mean a proper class of bananas? That's a lot of bananas.
Perhaps you mean what's called in computing a multiset, or (older terminology) a bag. If so say that. If you mean something else, say that. That's what notation is for: to engender clarity of communication.
So you mean they're identical collections, but their elements aren't equal? :S
{b, b, b, b, ... }
Is a multiset where 'b' has multiplicity=? I guess?
You are confusing me. Every element in both sets is indistinguishable and they are lined up with each other (in one-to-one correspondence):
{b, b, b, b, ... }
{b, b, b, b, ... }
We know all elements are identical 'b's and the collections both have the same infinite cardinality. The first elements are in one-to-one correspondence. If the nth element is in one-to-one correspondence, then so is the nth+1 element. So the two collections are surely equal?
Only those who (perhaps naively) embrace Cantor's mathematical model of a continuum as isomorphic to the real numbers would affirm this metaphysical implication, thus claiming that space is somehow composed of dimensionless points. Such a collection of distinct objects is a bottom-up construction in which the parts are ontologically prior to the whole. By contrast, I would argue that true continuity is a top-down construction in which the whole is ontologically prior to the parts. Positions are only actual if and when they are marked for a purpose, such as measurement; otherwise, they are strictly potential.
Assuming for the sake of argument space is a continuum (which I do not believe), when I move my hand from left to right, it would pass through an actual infinity of distinct positions. Every intermediate position is therefore actualised by motion and we are left with the conclusion that the particles in my hand passed through an actual infinity of intermediate positions.
I believe instead that each particle in my hand performs something like a quantum jump down at a microscopic level and that there are no true continua in our universe.
You are just choosing one (fringe) theory of cosmology; many theories do assume past time is actually infinite. See CCC by Roger Penrose as an example.
Quoting tim wood
I already gave you an example (Cantor). You need look no further than the philosophy forum for more examples: many folks on here argue that time has no start or that space is a continua, see https://thephilosophyforum.com/discussion/comment/367423 for example - and my response https://thephilosophyforum.com/discussion/comment/367425.
Lets look at what Cantor said again:
"Accordingly I distinguish an eternal uncreated infinity or absolutum which is due to God and his attributes, and a created infinity or transfinitum, which has to be used wherever in the created nature an actual infinity has to be noticed, for example, with respect to, according to my firm conviction, the actually infinite number of created individuals, in the universe as well as on our earth and, most probably, even in every arbitrarily small extended piece of space. - Georg Cantor
I've highlighted the relevant piece - he is talking about actual infinities in our universe.
Quoting tim wood
Yes, if you believe space is a continuum (which many people do), then simply moving your hand from left to right actualises an infinity of intermediate positions.
I'm thinking of them as sequences.
The first sequence, the one in your OP, is the sequence "the constant sequence where every element is b".
The second sequence, the second one in your OP, is the sequence "the sequence derived from the previous constant sequence by deleting every second element". This is also "the constant sequence where every element is b" since the sequence consists solely of the constant b... The constant is b, and every element is b.
The two sequences are identical. They also have the same cardinality. They are not identical because they're in a one-one correspondence, they're identical because all their elements are the same and in the same order.
- Infinite past time (leading to a belief that an actual infinity of moments has occurred in the past)
- Space is a continuum (leading to a belief that motion actualise an infinity of intermediate positions)
Not sure what else I can say here :sad:.
A function can have fixed points. These are when you get the same thing out as what you get in.
The operation "take a sequence, delete every second element, output the sequence without the deleted elements" is a function on the space of infinite sequences to the space of infinite sequences. It has a fixed point (produces an output equal to the input) whenever:
{a1, a2, a3, a4, a5, a6, a7, a8, ...}={a1, a3, a5, a7, ... }
a1=a1, a2=a3, a3=a5, a4=a7...
whenever all sequence elements are equal.
Notably, the sequence {1,2,3,4,5,6,...} of naturals produces the odds {1,3,5,...} upon applying this function. So, applying the function can produce the same thing (when the input sequence is a constant sequence), or produce a different thing (when the input sequence is not a constant sequence).
I think you are splitting hairs between 'believe' and 'aver'. Cantor states he has 'firm conviction' in the belief of the existence of the actually infinite. Aristotle believed/averred time had no start because:
- Time had no start because for any time, we can imagine an earlier time.
- Time had no start because everything in the world has a prior cause.
Newton held this belief also and many people still hold this belief. And such a belief leads to a belief in actual infinity. Similarly, the commonly held belief in space/time being a continuum again leads to a belief in actual infinity.
It is very convenient to regard the naturals, the reals as actually infinite sets, but this is merely a mental convenience to allow us to reason with all the naturals/reals. It does not mean that anything with the structure of the naturals/reals can exist in reality - they are a purely mental construct.
We can imagine all sorts of things in the mind - levitation, talking trees, actual infinity - but only a subset of what we can imagine is possible in reality.
I have a similar reaction to the axiom of choice. How can balls be selected from actually infinite bins? It is not possible to complete such a selection - the selection process goes on forever so it is impossible to complete. So again, we have something that is impossible in reality, but we can imagine it in our minds where the impossible is possible.
No, it would not. If space is truly continuous, then it is not composed of distinct positions. We arbitrarily impose distinct positions on space for various purposes, including measurement. They are entia rationis, creations of the mind, not constituents of reality itself.
There can be no infinite collections in reality because an infinite collection of something would leave no room for anything else in reality. The infinite collection would BE reality. Maybe reality is an infinite collection of space-time with finite amount of energy-matter.
Quoting Devans99
Are you saying that there is something beyond space and time? You're implying a boundary, but a boundary with what? Space and time are infinite. Energy/matter isn't.
I don't understand you. If my hand passes from position 0 to 1, then it passes through positions 1/2, 1/4, 1/8, 1/16, ... 1/?. So movement in a continuum actualises infinity, which is why I think everything must be discrete.
I think that there maybe pure nothing beyond spacetime - no time at all so nothing can be in any way whatsoever.
Time probably started 14 billion years ago. All expanding universes have a start in time and our universe has always been expanding.
Space has been expanding at a finite rate for 14 billion years so it must be finite in size.
I guess it comes down to whether you require that the application of a function to an input necessarily produces a distinct output (IE, no fixed points). Trying to avoid this blocks lots of useful things.
This is the same mistake that Zeno made. Positions 0 and 1 do not exist unless and until we arbitrarily mark them as such, and the same is true of any and all intermediate positions between them. We cannot proceed to mark an actual infinity of those; we can only hypothesize that there is a potential infinity of such positions in accordance with our arbitrary system of measurement--for example, the real numbers. Moreover, the physical distance from 0 to 1 does not matter--whether it is 1 kilometer, 1 meter, or 1 millimeter, the real numbers assign the same multitude of intermediate positions between them; likewise if we change our designation of the second position from 1 to 2 or any other value. All this demonstrates that real space is continuous; we only model it as discrete for practical purposes.
But the action of movement does mark positions 0 and 1 and all positions in-between. We know that our hand actually passed through all those positions in the past, so if space is a continuum then motion actualises an infinity of positions.
As pointed out in the OP, actual infinity is absurd, so therefore space is not a continuum.
No, it only marks positions 0 and 1; marking any individual intermediate positions would require their explicit designation, and we can only ever do that for a finite quantity of them.
Quoting Devans99
No, continuous motion is the reality, while discrete positions are creations of thought to facilitate describing the motion.
Quoting Devans99
No, treating space as continuous does not require an actual infinity of positions, only a potential infinity of positions.
If my hand moves from position 0 to 1, it is guaranteed to pass through positions 0.5, 0.25, 0.125, etc... Or are you saying it somehow skips over intermediate positions? That would be discrete movement.
Quoting aletheist
But movement is something that actually happened in the past - my hand in the past moved through all possible positions - so that must be an actual infinity of positions.
Beyond the boundary is nothingness IMO. Nothing cannot be actually infinite because it is nothing. If it is other universes then they cannot be actually infinite because it would lead to the absurdities referenced in the OP. Or see here for another example of the absurdity of actual infinity:
https://en.wikipedia.org/wiki/Ross–Littlewood_paradox
No, the only actual intermediate positions are the ones that we individually mark. There is a potential infinity of such positions, but we can only mark (and thereby actualize) a finite quantity of them. Again, continuous motion is the reality, while discrete positions are our invention.
Quoting Devans99
Given whatever path your hand actually followed in moving between its initial and final positions, it indeed moved through all possible intermediate positions along that particular path--again, a potential infinity, not an actual infinity. In order to describe that path, we would need to mark various intermediate positions and assign coordinates to them relative to the initial and final positions, taking the distance between them as our arbitrary unit of length. The more positions we mark and measure, the more accurate the resulting description of the motion--but it will never be perfectly accurate, since we will never be able to mark and measure all the possible positions; merely a finite quantity of them, which are the only actual positions.
https://en.wikipedia.org/wiki/Ross–Littlewood_paradox
https://en.wikipedia.org/wiki/Thomson%27s_lamp
Paradoxes indicate we have a wrong assumption somewhere, in this case, the assumption that it is possible to complete an actually infinite number of steps in a finite time is suspect. So I doubt that true continuity is possible.
He is obsessed with proving God, so he ain't in my Heaven :)
The wrong assumption in this case is that the true continuity of space would require your hand to complete an actually infinite number of steps by passing through an actually infinite number of intermediate positions. As I have explained repeatedly now, the only individual positions that exist are whatever finite quantity of them we explicitly mark. If you still want to insist that real space is discrete, then make your case, but please stop pretending that this particular objection to its continuity is valid.
If space is continuous then my hand moves through an actually infinite number of intermediate positions. But actual infinite leads to contradictions. So I doubt that it can exist.
Also there is an information based argument. True continua would be structurally identical no matter what the size. So a millimetre of space would have the same structure as a light year. Suggesting the same information content. That is hard to swallow.
As explained above, you can use mathematical induction to see that all the bananas in each sequence are in one-to-one correspondence. So for all finite numbers n, the bananas are in one-to-one correspondence. At 'actual infinity' are the bananas in one-to-one correspondence? I think not because actual infinity is not a well defined concept. It is an illogical concept that cannot occur in reality IMO.
I'm glad we are in agreement. I think though that not everyone agrees with us. For example, the bedrock of maths:
1. Axiom of infinity. It claims that the set of natural numbers exist. They exist in our minds where the impossible is possible, but there is nothing like it in reality, so maths should not claim 'they exist'.
2. Axiom of choice. It claims it is possible to choose balls from an infinite number of bags. In reality, one cannot complete an infinite task, so it is impossible to make the infinite selection of balls. Hence maths should not claim it is possible.
I believe that the universe is both finite and discrete (=free of actual infinities). Plenty of people disagree so I think there is still a discussion to be had.
False. Again, if space is a continuous whole, then it is not composed of individual and distinct positions.
This indicates a confusion between existence in mathematics and actuality in metaphysics. They are not synonymous or equivalent. Everything that "exists" in mathematics is merely logically possible, not actual.
Quoting Devans99
This indicates a confusion between logical possibility and metaphysical possibility. Again, they are not synonymous or equivalent. It is logically possible to choose balls from an infinite number of bags, even though it is not metaphysically possible; i.e., it is actually impossible.
In summary, mathematics is the science that draws necessary conclusions about strictly hypothetical states of affairs. That includes its application to infinity--never actual infinity, always potential infinity.
You are missing the point I'm making. I believe that the naturals and reals are purely mental constructs. They exist in our minds only (where the impossible is possible). They have the same status as talking trees and levitation - illogical/impossible things can exist in our minds but they cannot exist in reality.
In the instances of the axiom of infinity and axiom of choice, maths departs radically from reality and that departure leads other folks astray (physicists, cosmologists, philosophers). That is why I raised this thread.
If you disagree, then please give an example of something that has the structure of the naturals from nature.
An actual infinity of naturals (IE a set with a greater than any number of elements) is impossible.
Quoting aletheist
It is not logically possible to complete a task that has no end.
All of these are logically possible, just not metaphysically possible.
Quoting Devans99
No one is claiming otherwise. When mathematicians state that the natural numbers "exist," they are not thereby calling them an actual infinity, only a potential infinity.
Quoting Devans99
Incorrect--it is logically possible, just not metaphysically possible.
They are not logically possible as you can see from the argument in the OP - assuming that they are logically possible leads to a contradiction. Or if you don't like that argument, see:
https://en.wikipedia.org/wiki/Ross–Littlewood_paradox
So the logical assumption of the existence actual infinity leads to paradoxes/contradictions. Paradoxes/ Contradictions indicate a logical error has been made, in this case the assumption that actual infinity is a logical concept.
Quoting aletheist
A potential infinity is like a limit - something approaches but never actually reaches that limit. Actual infinity is equivalent to the claim that the natural numbers exist - the axiom of infinity says they actually exist - not potentially. That leads to the conclusion that there is a set that exists with a greater than any number of elements.
Quoting aletheist
It is not logically possible to reach the end of something that has no end.
You simply refuse to acknowledge the definitions of terms that others are employing, and thus consistently (and persistently) attack straw men. Actual impossibility does not entail logical impossibility. Mathematical existence is not metaphysical actuality. The infinity of the natural numbers is potential, not actual. Continuity of space does not require an actual infinity of distinct positions.
Question: Do you think that "5 is prime" is true? Or merely logically possible? Or a complete fiction made up by evil set theorists?
I think "5 is prime" presents a challenge to those who say that math isn't true. "Actual" as you put it. Is "5 is prime" actual or not? And if not, what is it?
But logical impossibility, which actual infinity is (as demonstrated by the paradoxes/contradictions) does imply actual impossibility.
The set of natural is defined in maths as an actual infinity:
'The axiom of Zermelo-Fraenkel set theory which asserts the existence of a set containing all the natural numbers' - http://mathworld.wolfram.com/AxiomofInfinity.html
A set containing all natural numbers exists both logically and actually... but that leads to logical contradictions... so such a set is not logically possible.
Quoting fishfry
Yes, given the standard mathematical definitions, the proposition that the number denoted by "5" possesses the character denoted by "prime" is true. Do you think that either of these terms denotes something actual?
No, it is defined as a potential infinity. One more time: mathematical existence does not entail metaphysical actuality. No one, except perhaps an extreme platonist, claims that there is an actual set containing all the natural numbers.
Cantor did claim actual infinity exists:
"Accordingly I distinguish an eternal uncreated infinity or absolutum which is due to God and his attributes, and a created infinity or transfinitum, which has to be used wherever in the created nature an actual infinity has to be noticed, for example, with respect to, according to my firm conviction, the actually infinite number of created individuals, in the universe as well as on our earth and, most probably, even in every arbitrarily small extended piece of space. - Georg Cantor
Sure, but in the quoted text, he did not claim that there is an actual set containing all the natural numbers. And his (incorrect, in our view) belief that there is an "actually infinite number of created individuals" does not somehow falsify all of his mathematical ideas about infinity.
Quoting Devans99
We can prescribe how you would logically go about constructing the set of all natural numbers, but we cannot actually carry that process out to its completion.
Quoting Devans99
Please provide an authoritative reference for the claim that "all sets are actual." Remember, mathematical existence does not entail metaphysical actuality.
Yes I do. That's why I can't so easily sign on to the proposition that math isn't true. Some parts of math are obviously true, or actual. It's easy enough to say that abstract math isn't true in the sense of physically true. But there are truths that aren't physical. "5 is prime" is one of them. The axiom of infinity is NOT one of them. The axiom of infinity is taken as an axiom in set theory but is not (as far as we know) true or even meaningful in the actual world. But "5 is prime" IS true in the actual world.
Is that a distinction you find meaningful? Or do you not make a distinction between "5 is prime" and the axiom of infinity?
He believed actual infinity is possible logically and in reality. And many people are still under that impression. What we are all taught at school - the Dedekind-Cantor continuum - a line is an actual infinite set of points - is an actual infinity.
Quoting aletheist
"In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right" - https://en.wikipedia.org/wiki/Set_(mathematics)
Something that has potential but not actual existence is not well defined.
In my view, nothing within mathematics is actual--again, it is the science that reasons necessarily about strictly hypothetical states of affairs--and truth has to do with what is real, not just what is actual. Specifically, a proposition is true if and only if its subjects denote real objects and its predicate signifies a real relation among those objects. The real is that which is as it is regardless of what any individual mind or finite group of minds thinks about it, while the actual is that which acts on and reacts with other things.
Quoting fishfry
We agree on this--the number 5 and the character of being prime are real, even though they are not actual.
Quoting fishfry
How so? Given my definitions above, perhaps what you mean is that "5 is prime" is true in the real world; in which case, again, we agree on this.
Okay, but you and I agree that this now-standard mathematical definition of a continuum is philosophically faulty; in my case, because I hold that a line is not composed of an actually infinite set of points. Instead, the continuous whole is ontologically prior to any discrete parts, which only become actual when we arbitrarily mark off a finite quantity of them for some purpose.
Quoting Devans99
Please provide an authoritative source for this claim, or just acknowledge that you made it up. Here is what "well-defined" means in this specific context.
By virtue of the procedure by which we could logically go about constructing the set of all natural numbers, it clearly qualifies as well-defined in the relevant sense. Even though there is a potential infinity of its members, such that we could never actually assemble the complete set, we can always easily determine whether any proposed candidate is or is not one of those members. 5 is, but 5.1 is not. 750,943,179,981,061 is, but a banana is not.
So, can you provide an example of something whose membership in the set of all natural numbers is ambiguous? That would be the only way to demonstrate that it is not well-defined.
The axiom of infinity:
[math] \exists \mathbf {I} \,(\emptyset \in \mathbf {I} \,\land \,\forall x\in \mathbf {I} \,(\,(x\cup \{x\})\in \mathbf {I} )).[/math]
I believe (I'm not a mathematician so forgive me if I'm wrong) this reads ‘there exists a set I for which the null set is a member of I and for all x belonging to I, x union the set formed by x also belongs to I’.
I do not believe this is a declaration of potential infinity - it is a declaration of actual infinity. It says there exists such a set - it does not reference limits or sequences or any of the mechanisms of potential infinity. It does not say such a set exists potentially - it says to me it actually exists.
I take your point that one could perhaps interpret it as potential infinity but I do not believe this is the common / mainstream interpretation.
Yet again: mathematical existence does not entail metaphysical actuality. Within mathematics, the number 10^100 (1 googol) indubitably exists and is a member of the set of natural numbers; but according to physics, the total quantity of actual particles in the entire universe is only about 10^80.
Quoting aletheist
I admit to not understanding the distinction between real and actual as you tried to explain it in this post. But consider Internet security. Are cryptocurrencies and online security actual? They have actual (in the everyday sense) impact in the real world of our lives. But online security is based on public key cryptography, which is 100% based on abstract number theory; namely, the theory of factoring large composite numbers.
It's interested that for over 2000 years, number theory was considered beautiful but useless mathematics. It wasn't till the mid 1980's that public key cryptography was invented and then used as the basis of all online encryption and security.
This is a striking example of abstract, meaningless math becoming suddenly actual.
What say you? Again I confess I don't know what you mean by real versus actual. Can you give an example?
I am largely employing the terminology and philosophy of Charles Sanders Peirce in all this, so I will offer a couple of his examples. If I were to hold a stone in my hand and then release it, then it would fall to the floor. This subjunctive conditional proposition represents a real law--one that is true regardless of whether I ever actually carry out the experiment that it describes. Similarly, any diamond really possesses the character of hardness, regardless of whether anyone ever actually scratches it with corundum to demonstrate that it does.
I do not know enough about public key cryptography to hazard a guess at how the distinction between reality and actuality applies to it. In general, I believe that the practical effectiveness of mathematics stems from its hypothetical nature; the key is that the idealized model must adequately capture the significant aspects of the actual situation. I am a structural engineer, so I routinely use a computer to simulate the effects of gravity, wind, earthquake, etc. on a building in order to design it such that it can be expected to remain standing once actually constructed.
Reminds me of Hilbert's Hotel. It helps to make a distinction here - that between quantitative and qualitative change.
Since your argument is just a reworked Hilbert's Hotel Paradox I'll go with the Hotel analogy.
Imagine a hotel with infinite rooms and with an infinite number of guests a1, a2, a3,... Now imagine a new guest b1 who wants to stay in that hotel. The manager simply moves a1 to a2, a2 to a3, aN to aN+1,... and b1 gets a1's room. Notice that though the quantity, infinity is still infinity, hasn't changed, the quality has: b1 is the new guest.
Similarly for an infinite number of new guests c1, c2, c3,..., we do the following:a1 moves to a2, a2 moves to a4, a3 moves to a6, aN moves to a2N,...which frees up the odd numbered rooms for the infinite guests c1, c2, c3,... As you'll notice though there's no change quantitatively there is a qualitative change, the guests c1, c2, c3,...are new.
Since infinity is an exclusively quantitative concept, it fails to register qualitative changes but that doesn't mean no change has occurred.
To illustrate take 3 people, Tom, Dick and Harry. Now imagine Tom gets replace by Jane. We still have 3 people viz. Jane, Dick and Harry. There's no change in quantity but there's a qualitative difference viz. Jane is now in place of Tom.
My argument uses sequences of identical bananas, so that the 'quantity' and 'quality' of bananas both are constant whilst bananas are added and removed from the sequences - resulting in absurdity.
Quoting aletheist
A googol-sized set could have logical existence if our universe was bigger. A greater than any number sized set does not even have logical existence (leads to absurdities so it cannot be logically sound).
By that reasoning, an infinite set could have logical existence if our universe was infinite. But logical possibility is not at all dependent on actuality, or even metaphysical possibility. So infinite sets do exist, in the strictly mathematical sense of existence.
Quoting Devans99
It only leads to absurdities if one insists on attempting to apply the same rules to infinite sets as to finite sets. Mathematicians have long recognized this, which is why there are different rules for infinite sets.
The same rules apply for finite and infinite sets. And if you don't like the absurdity in the OP, see this famous example for proof that actual infinity is logically absurd:
https://en.wikipedia.org/wiki/Ross–Littlewood_paradox
Why not avoid all the redundancy by putting b1 in the next room aN+1?
As long as you continue to insist on this, there is nothing more for us to discuss.
Surely this statement is a contradiction? Surely in our reality, when something is changed, it changes?
So I think we have to conclude that actual infinity is not part of our reality (or indeed any logical form of reality), it is just an illogical concept that exists in our minds (along with concepts like levitation, talking trees and square circles).
Interesting questions. I'm thinking the simple answer is that infinity is indeed part of our abstract reality. And that reality has contradiction and unresolved paradox.
Examples of abstract reality include the paradox of time, mathematics (Pi), cosmology (infinite universe theories), metaphysical phenomena, consciousness, so on and so forth..
Interesting discussion!
I think I see what you mean, maybe you can expand? I see that Pi cannot, in our reality, ever be actualised as it has infinite digits. A perfect circle cannot exist in our reality. But a perfect circle exists as an idea in the mind (along with talking trees). Is therefore a perfect circle an illogical/impossible idea or would you call it abstract reality?
I think the example of the axiom of choice is an interesting point. It claims it is possible to select one ball from an infinite number of bins:
1. Clearly in concrete reality, it is not possible to complete a never ending task
2. It also seems illogical for the mind to allow that we can complete a never ending task
3. Yet we can imagine it so and imagine the consequences. This is maybe the abstract reality you refer to?
There is of course much paradox/contradiction associated with language/self-reference, as you probably already know. Gödel's incompleteness/infinite theory speaks to that through both language and mathematics (liar's paradox and variations of same).
Also, that's a great question viz illogical or abstract. I wonder if it is simply an " illogical abstract " that actually exists in reality. Much like the metaphysical phenomenology of how the subconscious and conscious mind work together in an illogical manner (violating the laws of bivalence/LEM).
In a similar way, one answer to that question of abstract reality, I think, is the fascination with the conundrum of being and becoming. At the risk of redundancy from another thread, the paradox of time and your notion of infinity is very intriguing:
Quoting 3017amen
I think reality is strictly logical; I can find no instances of paradoxes/contradictions in reality. I think that the paradoxes/contradictions result from our models of reality rather than reality itself. Fundamentally our minds are capable of illogical reasoning but reality seems constrained to be logical only.
Our minds are part of a logical reality, but yet they are not constrained to purely logical concepts. When we think top-down about concepts, illogical concepts can surface in our mind. When we think bottom-up about concepts, these 'illusions' are often banished.
I think that 'abstract reality' might be a product of top-down thinking. It is easy to imagine that it is possible to square a circle when you think about it top-down. Yet it took 1000s of years of maths to prove it is logically impossible. So when we think about actual infinity, we are thinking about something in a top-down manner (sure something can go on forever). It is only when the concept of actual infinity is probed in detail that we see the problems - the paradoxes/contradictions show us that in fact we are imagining something illogical.
You physically can't keep putting 10 balls in a vase, while only removing one. It's an unrealistic thought experiment.
I'm not talking about any type of container holding an infinite number of universes. It doesn't make sense to talk about some container with an infinite number of things. Maybe that is your problem.
If causation (space-time) isn't infinite, then you have to come up with an explanation as to how something came from nothing. I think that is a much more difficult problem to deal with than conceiving of infinity.
Why is it unrealistic?
Quoting Harry Hindu
I don't believe everything came from nothing, I believe that something has permanent, atemporal existence and that something caused everything else. See here:
https://thephilosophyforum.com/discussion/7391/everything-in-time-has-a-cause/p1
On the contrary--if time were discrete, then it would necessarily consist of durationless instants at some fixed interval. The fact that "now" cannot have zero duration requires time to be continuous, such that "now" has a duration that is infinitesimal--not zero, yet less than any assignable or measurable value.
Discrete time would consist of discrete, non-zero, non-infinitesimal time slices - so they would have a duration.
On the other hand, if time was composed of infinitesimal time slices, then each fixed period of time would be composed of 1/?=0 length time segments giving a zero total length for all elapsed intervals. Which cannot be right.
In other words, continuous lapses of time with finite duration, arranged such that each one starts when the previous one ends. Calling this "discrete time" is a misnomer.
Quoting Devans99
The last sentence is correct, because the first sentence demonstrates a complete misunderstanding of infinitesimals.
They are distinct, like a movie plays at 60 frames a second, each frame a time slice. There is nothing continuous about that.
Quoting aletheist
OK, what is the infinite sum of every possible infinitesimal:
https://www.wolframalpha.com/input/?i=limit+%28n%2Finfinity%29+as+n-%3Einfinity
It's zero. So infinitesimals cannot be the constituents of time because all time intervals would have zero length and time would not flow from one moment to the next.
That is exactly what I described as discrete time, not what you described. Each individual frame of such a movie corresponds to an instantaneous state, with zero duration, arranged at a fixed and finite interval of 1/60th of a second. There is no flow of time between the frames, just a leap from one to the next.
Quoting Devans99
Again, this reflects a complete misunderstanding of infinitesimals. A moment of time has a duration that is not zero, but is less than any assignable or measurable value relative to any arbitrarily chosen unit. The present moment includes an infinitesimal portion of the past and an infinitesimal portion of the future, which is why time does flow from one moment to the next.
If you truly want to understand the mathematics of infinitesimals, I recommend learning about synthetic differential geometry, also called smooth infinitesimal analysis. These links are to excellent brief introductions by Sergio Fabi and John Bell, respectively.
Infinitesimals are deeply illogical/impossible concepts and are shunned by most of maths. As demonstrated in the op, ? leads to logical absurdities, so logically 1/? must be absurd too.
There is nothing inherently contradictory about the mathematical concept of an infinitesimal, which is not necessarily defined as 1/?. Again, if you truly want to understand, please read one or both of the short articles that I linked. If you prefer to remain ignorant, carry on.
https://en.wikipedia.org/wiki/Hyperreal_number
The hyperreals are a model of the first-order axioms of the real numbers that contain infinitesimals. To construct them requires a weak form of the axiom of choice. In the 1970's nonstandard analysis, based on the hyperreals, was touted as a better way to teach calculus. Results are mixed. Most studies showed that students come out of standard or nonstandard calculus classes equally confused. No new math is introduced with the hyperreals; that is, anything you can prove with them you can already prove with the standard reals. That's why they haven't caught on. The standard reals have extensive mindshare. But infinitesimals have been made logically legit as of 1948.
Quoting aletheist
Yes, yet another way to do infinitesimals. I'll check out your links since I don't know much about SIA.
Quoting aletheist
Quibble I must. Nobody knows what a "moment of time" is, or if it's modeled by any of the various mathematical models of infinitesimals. I object to this claim. There's no currently accepted theory of physics that supports what you said AFAIK. Math [math]\neq[/math] Physics.
I have come across some arguments that teaching calculus using SDG/SIA--grounded in category theory, rather than set theory--is more effective than either standard or non-standard approaches. In any case, scholars of Peirce's mathematical thought seem to agree that it comes much closer to being a rigorous implementation of his conceptions of infinitesimals and continuity than NSA. For one thing, the principle of excluded middle does not apply--i.e., the logic of SDG/SIA is intuitionistic, rather than classical--which is exactly what he consistently maintained about anything that is general, including anything that is truly continuous. For another, functions in SDG/SIA "are differentiable arbitrarily many times" (Bell), consistent with this statement.
So just as I call modern constructivism Brouwers revenge, I can call SIA Peirce's revenge. This is very interesting. Do you happen to know how SIA relates to constructive math? They both deny excluded middle as I understand it.
What's R 300? Where can I find Peirce talking about calculus? I don't suppose I could ask for a simple explanation of what this phrase means? Is it anything like the quotient of dy and dx being the derivative when dy and dx "become" zero but aren't actually zero, as Newton thought of it?
ps -- Why are there so many Peirceans on this forum? I never hear about him anywhere else but his ideas are incredibly interesting.
Quoting aletheist
Aha. That's also true of the complex analytic functions in standard math. If a complex function (function from the complex to the complex numbers) is differentiable once, it's automatically infinitely differentiable. So complex differentiable functions are extremely well-behaved. Whereas real number functions (reals to reals) can be very wild, and differentiable only once, or twice, or some finite number of times before they are no longer differentiable. I wonder if this is related to SIA somehow.
Heh, I like it! I am an engineer, not a mathematician, so I would welcome your thoughts on SDG/SIA--although they probably belong in a new thread.
Quoting fishfry
Not really, but I suspect that they have different reasons for denying excluded middle. You might be interested in John Bell's book, The Continuous and the Infinitesimal in Mathematics and Philosophy. The first half covers the history, while the second half consists of chapters specifically on topology, category/topos theory, NSA, constructivism/intuitionism, and SDG/SIA. A new version just came out with an even longer title, but best I can tell the only significant change is the addition of several appendices on various topics.
Quoting fishfry
R 300 means manuscript number 300 as cataloged by Richard S. Robin in the 1960s. That particular text is incomplete and largely unpublished, but there is a transcription online. Peirce does not actually talk about calculus much in it, and I honestly do not know where else in his voluminous writings he might have done so in any detail. You might find some leads in the Robin catalog, and then you can browse through images of the actual manuscripts.
Quoting fishfry
Here is what Peirce wrote right before the sentence that I quoted.
Presumably the "false continuity" that he had in mind was that of Cantor. Does this help at all?
Quoting fishfry
I am not sure that there are really so many of us here, but I obviously agree with that last part. The problem is that Peirce never wrote a magnum opus spelling out his entire philosophical system, or for that matter any significant portion of it.
The Collected Papers (eight volumes) are arranged topically, rather than chronologically, with some rather misleading results because his thought definitely evolved over time. Unfortunately, the comprehensive critical edition being prepared by the Peirce Edition Project has been stuck for ten years after producing only eight volumes of thirtyish planned, primarily due to chronic lack of funding. The Essential Peirce (two volumes) is probably the most accessible compilation, but emphasizes general philosophy. Two specifically mathematical compilations--Carolyn Eisele's New Elements of Mathematics (four volumes in five) and Matthew E. Moore's Philosophy of Mathematics: Selected Writings--are probably more relevant to you.
I don't know enough. I know that it's categorical in flavor ... but then again so is differential geometry. I'll dispatch a clone to study up on SIA and another clone to read the collected works of Peirce ... uh oh I haven't got any clones. That means these tasks might not get done any time soon if ever.
Sounds like Peirce is rejecting discontinuous functions. Or something like that. I know that in constructive math, all functions are computably continuous or something like that. Makes some of the problems go away.
Quoting aletheist
Oh jeez I gave that link a look. Not exactly a clear writer IMO. I don't think I could ever slog through this.
Quoting aletheist
Dispatching a clone. Not the first time I've had this book recommended to me. Sigh.
Quoting jgill
I always thought this was an abstraction of basic 17th century calculus, where higher powers of infinitesimals can be ignored.
The problem with the articles you linked is that in both cases, a wrong assumption is made at the start of discourse:
"In order for SDG to be consistent, the law of exclude middle must not hold. SDG does not rely on classical logic but on intuitionistic logic."
I happen to strongly believe that the LEM holds for our universe and indeed all possible universes.
"Now if it were possible to take ?x so small (but not demonstrably identical with 0 that (?x)^2 = 0"
I also strongly believe there is no nonzero x such that x^2=0.
Why should I invest time and effort learning subjects that are based on wrong assumptions? All 'knowledge' I'd acquire in doing so would be inherently unsound.
I explained why. Pay attention. What vase would be large enough to keep putting in 10 balls while only removing one? The vase would have to be an infinite sized container, which makes no sense. How can something be both infinite and contain?
Quoting Devans99
If it has an atemporal existence then that is the same as saying that it doesn't exist. How does something cause everything else without being in time itself? How does it cause anything without changing itself? Even God has to exist in time if God changes. Change is time.
There are no "wrong" assumptions in pure mathematics. It is the science of reasoning necessarily about hypothetical states of things.
Quoting Devans99
LEM holds for anything that is determinate, including anything that is discrete; but it does not hold for anything that is indeterminate, including anything that is truly continuous. There is nothing illegitimate about intuitionistic logic.
Quoting Devans99
What you believe is irrelevant. There is nothing self-contradictory about defining an infinitesimal as that which is not itself equal to zero, but whose squares and higher powers are equal to zero; and it turns out to be quite useful.
Quoting Devans99
The problem is treating assumptions other than your own as indubitably wrong and refusing even to entertain them, which is a textbook example of sheer dogmatism. Unless you consider yourself to be infallible, you might want to try opening your mind a bit.
Good point. Time is eternity; eternity time.
I suppose that's one of the vexing problems in making sense of an unchanging Being or thing, in a world of change. Time seems to be an abstract reality, yet no different than other abstract realities that exist and are perceived in consciousness [conscious existence] like the existence of mathematics itself, etc..
I know what you mean about clones! Hopefully you can at least digest the two short articles about SDG and SIA. Browsing Moore's single volume of Peirce's writings on philosophy of mathematics might be enough to give you the gist of his overall approach and help you decide whether delving deeper is worth the trouble.
Fair point, but assumptions that stray wildly from common sense / common experience indicate the subject is squarely pure rather than applied maths. IE it tells us nothing about our reality, it is telling us something about an alternative reality where common sense does not apply. I am not interested in maths for maths sake, I am interested in what it tells us about the reality we live in. If parts of maths adopt axioms that depart from common sense, then I have to disregard those parts when searching for a description of our reality.
This is the way you should approach maths (and other fields of human knowledge) - you look at the axioms and decide if you believe them or not. Then you learn about the parts that you believe have sound axioms and disregard the rest. Parts of math claim that for non-zero x, x^2>0. Other parts of maths claim that for non-zero x, x^2=0. The two parts of math are incompatible. It is only possible to hold a belief in one of these two incompatible parts of math. I put my money on basic arithmetic.
Common sense tells us that common sense is highly fallible. Some developments in mathematics and science over the centuries are highly counterintuitive, and if we had insisted on sticking with common sense, we would still be misunderstanding reality. Is it common sense that there are numbers incapable of being calculated as fractions of integers? Or that matter consists of atoms that in turn consist of varying quantities of protons, neutrons, and electrons? Or that gravity is the curvature of spacetime, rather than a direct force of attraction between massive bodies?
Quoting Devans99
Nonsense, there is no single set of mathematical assumptions that perfectly matches reality--just different models that are useful for different purposes.
The paradox starts with the assumption that actual infinity is possible, so it is OK to assume an actually infinite bag/vase.
Quoting Harry Hindu
I have done a probability analysis of all the arguments and I get 94% certain that time has a start.
That implies something atemporal must very probably exist in order to be the cause of time. I imagine the whole of the universe as a 2d spacetime diagram of finite size. Then I imagine the atemporal thing (God) off to the side (not on the plane) and a mapping between the atemporal thing and each point in the plane. Then the atemporal thing can express itself in spacetime without being part of spacetime.
I am not sure precisely how the atemporal thing (God) could work. But then if you think about all the universes in the multiverse, all the multiverses in reality and all of the different possible realities that might exist, it seems impossible that we would ever understand them all - so things with a drastically different nature very probably exist - including atemporal things.
Time enables change. Time is not change. If time was change then time would flow faster in the presence of change, yet SR indicates time slows down in the presence of change.
Who knows. I don't disregard non-euclidean geometry because its axiom that parallel lines meet has a possibility of being true. I do however disregard maths that does not follow the LEM. There is so much maths, all different fields with different axioms that disagree with each other - it is therefore required to be selective in what one chooses when trying to use maths to understand the universe.
The proof of irrational numbers is common sense. We have empirical evidence for atoms and the curvature of spacetime. So these things are in agreement with common sense.
Quoting aletheist
But as seekers of a truthful explanation of our reality, we have to make choices between incompatible branches of mathematics. I'm unwilling to discard arithmetic from my set of choices of valid mathematics.
We clearly have very different definitions of "common sense."
Quoting Devans99
Who said anything about discarding arithmetic? It is very useful for very many purposes, especially those encountered in everyday life, which generally involve dealing with finite quantities of discrete things. A different approach is required to handle potentially infinite sets, and yet another is required for true continuity. Whether this accords with "common sense" or not, it is the reality.
Your chosen version of reality included continua (incompatible with arithmetic IMO) and infinitesimals (incompatible with arithmetic IMO). My chosen version of reality does not include these two concepts (I think reality is finite and discrete). Time will tell which version of reality is correct.
No one gets to choose a "version of reality," because by definition reality is as it is regardless of what anyone thinks about it. Adopting finite discrete arithmetic as the sole or primary basis for your entire metaphysics strikes me as extremely naive. I hope that someday you will be more open to alternatives.
Unchanging causing change is as incoherent as something coming from nothing.
Quoting Devans99
See? You can't escape talking about God relative to the universe. You are implying space-time encompassing God and your universe, as God is located relative to the universe and expresses itself in time.Quoting Devans99
They exist only as imaginings in the human mind in this particular universe.
Quoting Devans99
I dont know what you're talking about. Maybe you're talking about realtive change. There is more or less change in one area relative to another.
I did not say timeless beings cannot cause change. I said that a timeless being can express itself in spacetime and thereby be the agent of change.
Quoting Harry Hindu
I am not implying space-time encompassing God, I said God is external to spacetime but can express himself in spacetime.
Spacetime is fine-tuned for life. That requires a fine-tuner from beyond spacetime. Nothing can exist forever in time, that requires a first cause from beyond time.
Quoting Harry Hindu
You cannot possibly have understanding of every possible reality beyond our own, so you cannot make such a claim.
Quoting Harry Hindu
What I mean is that special relativity says that time is observed to slow down as things move faster (hence the photon moving at the speed of light is a timeless particle). So more change = less time. So time is not change (because that would lead to more change = more time - in opposition to special relativity).
True enough. In computations involving non-infinitesimal calculus higher order terms can be ignored depending on the settings.
Quoting fishfry
Hmmm. I keep learning things here. Thanks. :chin:
https://people.eecs.berkeley.edu/~fateman/papers/limit.pdf
Well then you're contradicting yourself. Things can change either qualitatively or quantitatively and you say neither has occurred. Then in what way have the sets changed; after all your claim is that when it is changed, it is not changed.
The argument in the OP is that you can add/remove identical items to an infinite sequence and the sequence remains identical/unchanged (both qualitatively and quantitatively). This results in the contradiction 'when it is changed, it is not changed', which is what I intended - assumption of the existence of actual infinity leads to a contradiction.
So add/remove is the change. How? In what way have you changed the infinite set from which something has been removed and the infinite set to which something has been added? You deny that the change is qualitative since you claim you're using identical bananas. You deny that it is a quantitative change since infinity + 1 = infinity and infinity/2 = infinity. So, it must be that nothing has changed and that precludes any claim that the sets have changed in any way.
Is change that is neither qualitative nor quantitative possible?
Also I have an issue of identicalness of the bananas. How does this identicalness weigh in on the issue? Since you've listed your sets as {b, b, b,...} it implies that there's a difference between any two b's; after all if they were logically identical in that all b's refer to one and only one object then you wouldn't or rather couldn't list them separately as {b, b, b,...}; set theory doesn't allow repetitions of elements. Since the difference between b's isn't quantitative because b corresponds to the number 1 it follows that the b's differ qualitatively and that points to a qualitative change when you manipulate the two sets as you do.
We add 1 banana to the sequence (=it should change quantitatively and qualitatively).
But is does not change quantitatively(?+1=?) or qualitatively(still identical rows of identical bananas).
So there is something that when we change it, it does not change
That's a contradiction so our premise must be wrong
Hence actual infinity is impossible.
Quoting TheMadFool
Thats why they are sequences rather than sets (sequences allow duplicate objects).
It does change. The problem is in your definition of identicalness.
1. Logical identicalness. I'll use examples to make it clear.
Charles Lutwidge Dodgson is identical to Lewis Carroll. There's only ONE object but with different names. Carl Lutwidge Dodgson and Lewis Carroll can occupy the same space at the same time. There's absolutely no difference between them.
2. Xerox identicalness. Identical twins or two instances of the same car model. Identical twins or two instances of a car model cannot occupy the same space at the same time. There's a difference there.
Your bananas are not type 1 identical because then there would be only ONE banana. Ergo, your bananas are type 2 identical but then there's a difference between each instance of such identicalness by virtue of their inability to occupy the same space at the same time. It's this difference that produces the change in your two sets.
Each banana has a different spatial position I agree, but the two sequences, ignoring their space time position are identical (same mass, same number of bananas, all bananas in one-to-one correspondence). The definition of a sequence (similar to a set) does not include their relative spacial positions - so the two sequences of bananas remain identical whilst they are changed. Which is a contradiction. Hence actual infinity cannot exist.
You can't ignore their space-time positions because it's critical to your argument. Why are there infinite bananas? Because they occupy different spaces? If they occupy the same space there would be only one banana.
I think the two sequences are identical in that if there is a banana at spacial position 1 in the first sequence, then there is also an identical banana at spacial position 1 in the second sequence.
If you don't buy my proof via contradiction that actual infinity is impossible, what about a proof via reductio ad absurdum:
https://en.wikipedia.org/wiki/Ross–Littlewood_paradox
So we have an infinite bag and we add ten balls and remove one. We repeat that an actually infinite number of time. At each finite step, there are 9n balls in the bag. At actual infinity, there are zero balls in the bag. Reductio ad absurdum, actual infinity is impossible.
And those parts are discrete and finite:
1. The parts can't be size zero or size undefined as then they could not constitute the whole
2. The parts can't be size 1/? because they would not constitute the whole and because ? leads to contradictions
3. So the parts must have a finite, non-zero size. IE discrete
You need to understand the difference between what is possible in your mind (where sure you can go on dividing forever and trees can also talk) and what is possible in reality.
step 1, 9 balls numbered 2, 3, 4, 5, 6, 7, 8, 9, 10
step 2, 18 balls numbered 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
step n, 9n balls numbered n+1, n+2, n+3,...10n
The bag isn't empty because at the nth step even if the nth ball is removed, the balls numbered n+1, n+2,...10n are still in the bag.
Mathematical induction leads to 9n balls in the bag at each finite step (where n belongs to the natural numbers). You can't use mathematical induction for the infinite part of this problem as it applies for all n belonging to the natural numbers only and ? is not a natural number.
You need to understand that what the mind thinks geometrically of an object actually applies to it. How many parts does a banana ACTUALLY have? Don't say one because I can split it in half. And if I was all-powerful I could split it up infinitely. Objects are both infinite and finite at the same time. Logic proves this
No it does not. My mind thinks of levitating dogs on a regular basis. No dogs levitate in reality. The mind is fundamentally illogical so the impossible is possible in the mind (with the aid of fuzzy, top-down, thinking). What is possible in reality is a completely different question from what is possible in the mind.
Quoting Gregory
If you were all powerful you could not split up a banana indefinitely - you would never finish the process so its impossible.
If we grant you were beyond time (timeless), you'd still need a hypothetical continuous substance to subdivide. But no such substance is possible as I pointed out here:
https://thephilosophyforum.com/discussion/comment/368770
Continua are mathematically impossible to define so in all likelihood, they do not exist in reality.
Discreteness does even mean anything. Does the discrete have parts? If not it's zero and has nothing to do with an object
Fundamentally everything is composed of parts and the parts cannot be zero sized or infinitely small. So they must have non-zero finite size. This has been our experience with matter (molecules, atoms, quarks etc...).
Something discrete is a part and it does not have any sub-parts. It is indivisible. A pixel on your computer screen is an imprecise analogy. In reality it is a particle - a packet of discrete energy, probably taking on a wave form. It is not dividable into subparts.
"everything is composed of parts and.. must have non-zero finite size."
Then you keep thinking of the division. The numbers of parts go on forever. So infinity does exist
Do you potentially have a hand, or do you actually have one? How can something have parts only potentially? How can something exist yet not have parts? These are all non-sensical statements.
Just check the math. In the 1st step the 1st ball is removed but there are more than 1 ball. In the 2nd step the 2nd ball is removed but there are more than 2 balls. Ergo at the nth step then nth ball is removed but there are more than n balls.
? leads to contradictions so cannot exist. So neither can the inverse (1/?) exist.
Quoting Gregory
I think that my hand exists in actuality and is composed of discrete parts that move through spacetime in discrete steps.
That applies for all n belonging to the natural numbers. But the proof is about what happens at the point of actual infinity, which is not a natural number. The proof is all about showing that actual infinity is impossible.
Does the discrete part have parts. If it doesn't, why isn't it zero?
I think it is likely that reality has a nature akin to a computer screen made out of pixels. So each discrete part has a size of 1 (say). It makes sense then (in the mind) to talk of size 1/2, but such cannot exists in reality; it is merely a mental construct (in the mind, the impossible is possible).
The world is physical, which is made of infinite parts. If it's more like a simulation, than why are you elsewhere arguing for a God?
I think the world is made of finite parts; an actual infinity of anything is impossible; see the argument in the OP. What would an actual infinity of things be? It would be a set with greater than any number of elements - nonsensical - such an aberration can only exist in our minds, where the impossible is possible.
I doubt it is a simulation, but if it is, I believe God is the ultimate cause of that simulation.
Are the parts non-zero? Do they have a front and back? Uh, the front and back are parts! This is the paradox started by Zeno. YOU don't have the solution
Does a quark have a front or back? You can imagine it having so in your mind, but in reality it is an indivisible whole and it is not possible to address its front or back; only the whole unit.
A valid solution to Zeno's paradoxes is the universe is discrete and actual infinity does not exist.
It either has a back and front, or it doesn't. That is, it is either real or zero
How do you tell the difference between front and back? You could fire photons at the discrete object. But the minimum wavelength of a photon is much larger than the object. So you cannot detect front or back. Front or back are concepts that exist in your mind due to your experience with everyday macro objects. The microscopic world is different and quarks do not have fronts or backs. You can only imagine front and back of a quark because your mind (incorrectly) associates macro level attributes to micro level objects.
"discrete object" dont exist. Or maybe the do, but they are nonsensical.
What about space? Is space finite? What is there outside of space? Nothing/something. The former is space and the latter requires space.
Your premise of infinity lacks one major factor. That is, continuity.
Infinite also means non-stop (endless). Even before the addition and division which you mention, the collections must have been constantly progressing in size and, possibly, in as many progressions as is possible e.g. arithmetic, geometric, logarithmic, exponential, etc.
Mathematics works with defined limits, so when it encounters an undefinable 'process' (theoretically), it shows how immune it (the undefinable process) is to any of its defined (limited) operations.
I believe that spacetime started expanding 14 billion years ago at a finite rate, so hence spacetime must be finite currently. I believe there would be pure nothing beyond the boundaries of spacetime - there being no time or space for anything to exist. Pure nothing has no dimensions so it cannot be infinite.
I think you are referring to potential infinity? I don't deny the existence of potential infinities; the argument in the OP is squarely aimed at actual infinity.
From what I get outside of mathematics, the word infinite set or set of infinite 'anything' is an oxymoron. Because infinity or an infinity of anything contradicts with the meaning of set, which implies a kind of definiteness (and a kind of limitation). However, infinite set as used in mathematics is a way of expressing a concept which isn't realizable (factual).
That means, infinity is a concept, whether actual, potential or other.
None of this says anything about the world we find ourselves in, only our rational response to it. A peculiarity of our own mind. So I agree with the OP.
I agree they are concepts, not numbers. Focusing on actual infinity only, it is an illogical concept that can only have existence in our minds. If we assume actual infinity is possible then it leads to contradictions (see the OP) and absurdities (Hilbert's hotel etc...).
Quoting Punshhh
Indeed, I think that actual infinity is the product of top-down thinking. Bottom-up thinking shows it leads to absurdities. So with top-down thinking, illogical things are possible in our mind. It only becomes apparent that they are illogical through bottom-up thinking - and illogical things cannot have existence in reality. An example is that top-down thinking suggests we can construct a square with the same area as a circle (using ruler and compass). But 1000s of years of bottom-up thinking has finally proved that is impossible.
It does not really tell us anything about actual infinity - they merely assume such a quantity (and its inverse) exist. It may have applications but telling us about the nature of actual infinity is not one of them.
There is so much math I could learn and life is too short to learn it all. So I choose to learn the areas of maths that are based on sound axioms and disregard areas that are based on unsound axioms.
Holding an unwavering position on those principles is unwise. There is much paradox in this world and you are not going to get far being narrow-minded like you are (on this subject and the God one too). You know what you want to believe and are using all your mental effort to defend. My mind is like water. I let it go wherever truth pulls it. I don't have any structure whatsoever behind it saying "this is what I want to believe".
Like Shandy's diary, a veridical paradox, i.e. counterintuitive, yet does not otherwise derive a contradiction.
We change it in one respect (whether it includes this particular individual member), but it is not changed in another respect (its cardinality as an infinite set). Not a contradiction.
We change the sequence (its a sequence of identical bananas; not a set) to include an extra identical banana and we get back an identical sequence of identical bananas. So both the sequence itself and the cardinality remain unchanged despite us adding one banana.
The only basis for claiming that the two infinite sequences are "identical" initially is that they allegedly consist of "identical" bananas in "identical" order. Accordingly, adding another "identical" banana to the beginning of one of them is not really a change, since the new "first" banana is indistinguishable from any other.
I think you are trying to defend the indefensible; actual infinity is a logical impossibility. For example, is it possible to construct actual infinity or its inverse mathematically or otherwise? No it is not, each is a task that never ends and never ending tasks are impossible to complete.
Something that goes on forever like actual infinity would just be pure magic and magic is not possible because it defies logic and reality is logical. Is there anything with the structure of the natural numbers in reality? No so aleph-zero is just the invention of a deranged mind.
What are aleph-zero, aleph-one? They are names of patterns formed by imagining the abstract and illogical structure we call infinity. They are not sizes or cardinalities; just names of patterns or organisations. Does it make sense to add one to a snowflake? Or multiply a snowflake by a hexagon? Cantor had it all wrong about set theory just like he had it wrong about God talking to him.
Again, we have changed it in one respect but not in another - no contradiction.
Quoting Devans99
We are discussing hypothetical infinity, not actual infinity. We do not have an actually infinite sequence of actually identical bananas, let alone two such sequences.
If I add one banana to the sequence, I should get back something that is someway different from the original sequence. The fact that I get the same sequence back is a contradiction.
You seem to be saying it is possible logically and/or in reality to change something and it does not change. Your logic / reality therefore differs from mine.
Quoting aletheist
Can you explain the difference between hypothetical and actual infinity?
Not when all the bananas are stipulated as identical.
Quoting Devans99
It is possible to change something in one respect without changing it in another respect. If I peel a banana, it is still the same banana, even though I have changed it in one respect.
Quoting Devans99
I honestly do not see what there is to explain. Do you not know the difference between the hypothetical and the actual?
But the mass of the sequence must have changed. But the maths says the sequence is identical so it has the same mass. So its a contradiction.
Quoting aletheist
A peeled banana is no longer identical to a non-peeled banana.
Quoting aletheist
I assume by hypothetical you mean the imaginary structure of actual infinity in our minds? But hypothetical means it might or might not be true. In this case, it cannot be true. It is impossible to actualise infinity in the mind; just dreaming illogically about it is as close as we can get.
I think of actual infinity as both a logical concept and something that could apply to reality. But it is illogical so it can exist only in our minds (where the impossible is possible) and reality is logical so it cannot exist in reality.
If you don't like my proof actual infinity is impossible, how about this reductio ad absurdum proof:
https://en.wikipedia.org/wiki/Ross–Littlewood_paradox
So there are 9n balls in the bag for each finite step. There are 0 balls in the bag after actually infinite steps. An infinite number of steps is just the sum of the actions of the finite steps and none of the finite steps result in 0 balls in the bag. Reductio ad absurdum. Actual infinity is logically impossible.
A sequence has no mass, since it is a mathematical concept, not anything physical. An actual collection of bananas would have mass, but it would necessarily be finite, such that adding a banana would indeed add mass.
Quoting Devans99
You have made it clear that you reject the established mathematics of (hypothetical) infinite collections, but please stop pretending that there is no such mathematics, or that it cannot be different from the more familiar mathematics of (actual) finite collections.
Quoting Devans99
It is no longer identical in one respect--whether it is peeled--but it is still identical in others. For example, it is still a banana, and most people would even say that it is still the same banana.
Quoting Devans99
"Imaginary structure of actual infinity"? Now that is a contradiction.
Quoting Devans99
Hypothetical means logically possible, not necessarily true or even metaphysically possible. A hypothetically infinite collection or sequence is logically possible, while an actually infinite collection or sequence is metaphysically impossible.