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I am becoming increasingly convinced that Wildberger's views are, if a little bizarre, at least not hopelessly inconsistent.

When I was reading the comments in the video following (MF17), somebody said something that shocked me a bit, because I was unable to give a rebuttal that I found satisfactory:

The reason I consider [the axiom of infinity] restrictive is that it forces you to accept the existence of a certain set that has all sorts of bizarre properties, like the existence of one-to-one and onto mappings from some sets to proper supersets. That screws up your attempts to assign cardinalities to sets (see the continuum hypothesis), and it doesn't even buy you much: can you have a set of all sets? Why not? The real question is, what do you think infinite sets actually buy you in terms of reasoning power?

The point about the continuum hypothesis is arguably meaningless; as I understand it CH says that it is hard to assign to each cardinality a set, but nothing about the difficulty of assigning each set a cardinality.

However, the core of the question is a bit harder for me to answer. My immediate response is it that allowing infinity gives a sort of completeness. But as the person pointed out, it doesn't finish the job, it just (dramatically) kicks down the line where the new notion of "too big" is. Introducing classes pushes it even further, but the same problem arises, I think: there is still no class of all classes.

So my next idea was, well, infinity doesn't buy me reasoning power but it does provide a satisfying source of many examples. But I'm not sure if that's true either: Wildberger's alternative to ZFC is (if consistent) a type theory, or at least it uses the language of type theory. I know very little about type theory so if you want to reference it in an answer, it would be great if you could use small words :)

With type theory on your side it's not even clear that I have a much less restrictive universe of objects which I can speak about; just a not-entirely-arbitrary boundary where sets are no longer permitted and types must take over. This could be dramatically the wrong picture, since as I said I am very new at this.

And now I'm stuck. Can anyone save Cantor's paradise for me?

Eric Stucky
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    Are you aware that are theorems about finite combinatorial objects whose proofs require infinite sets? (hence there can be no "comprehensive, self-contained discipline of finite combinatorial mathematics"). For example, see this answer. – Bill Dubuque Dec 25 '13 at 01:22
  • Nope, I was not. As close as I got was that I knew the finite Ramsey theorem could be proven from the infinite one. But I also know there is a finite (finitistic?) proof of the Ramsey Theorem. Thank you for bringing that to my attention; would you consider leaving it as an answer? – Eric Stucky Dec 25 '13 at 01:25
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    I feel like Professor Wildberger is like the man who can't understand the reason for art, asking why abstract paintings are useful, when we have perfectly good methods for painting what we see with our own eyes. Unintuitive and bizarre results? That's why I do it! –  Dec 25 '13 at 02:06
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    @Jason Pollock, is it? :-) – Asaf Karagila Dec 25 '13 at 02:27
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    @AsafKaragila: Hilarious! :) –  Dec 25 '13 at 03:38
  • I feel like the person quoted doesn't quite get some things. Like how the set of sets is a completely different issue from infinity. Personally, my main argument in favor of set theory (especially with infinity) is that it has an incredible wealth of pathologies. The more things it's capable of expressing that no one would possibly think of, the better. I'm always against alternative foundations that would keep its expressive power within the confines of present mathematical practice (much less a small portion thereof). – Malice Vidrine Dec 25 '13 at 04:36
  • The axiom of infinity squares pretty well with my notion of the set of natural numbers. I'd say having the natural numbers as part of my foundational machinery is buying rather a lot, and well worth the price. – MJD Dec 25 '13 at 04:38
  • @MJD: Even Wildberger's crazy views do not reject the notion of natural numbers; rather they reject them as having the same status of set as finite sets. Is it having them as sets, as opposed to types, really such a huge advantage? I imagine the answer is yes, but as I said in the question, I have no idea why. If you have a better picture of it than I do, please post an answer :) – Eric Stucky Dec 25 '13 at 05:50
  • @BillDubuque Eric Stucky didn't ping you, but I also think a variant of your comment would be a very good answer. – Mark S. Dec 28 '13 at 21:47
  • In fact, it would probably be the best answer :) – Eric Stucky Dec 28 '13 at 22:23
  • For Wildberger and most fellow cranks, it is mostly about the words and not the meaning. If you state $\forall n \in \mathbb{N} : 2^n \geq 2n,$ then they are typically repulsed. If you state the entirely equivalent "if $n$ is a natural number, then $2^n \geq 2n$", then they have no reaction, or at least they do not know how to attack you, except by starting on their own a rant about sets anyway. – Tommy R. Jensen Aug 11 '19 at 13:34

3 Answers3

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I could argue against Wildeberger's philosophical views directly, but I don't really see the point.

The reason is that philosophical arguments aren't a very good method for making decisions. What works much better, in almost every case, is experimentation and evidence. If we want to figure out the best foundations for mathematics, the only effective method is to experiment: we should invent some foundations, and then figure out what mathematics you can make with it.

So what does the axiom of infinity buy you? Well, all of modern mathematics.

I can't think of a single field of mathematics that doesn't use the axiom of infinity on a regular basis, and I can't imagine how virtually any of the important results in mathematics discovered over the last century could be proven or even stated without the axiom of infinity. It is enormously conceptually helpful to be able to consider infinite collections of objects.

Overall, ZFC has been quite successful as a foundation for mathematics over the last century, and it would take an enormous amount of evidence to convince the mathematical community to switch to some other foundation.

If Wildberger wants to show that mathematics would be better off without the axiom of infinity, he will need demonstrate that his alternative foundations are either cleaner or more powerful than ZFC. The first step would be to get some set theorists interested in a workable version of set theory that doesn't include the axiom of infinity, so that they can start to investigate the mathematics that results. This is not unheard of: there has been a lot of work, for example, on the new foundations as a possible alternative to ZFC, or on non-standard analysis as a possible alternative to analysis.

But it does very little good to argue abstractly about possible shortcomings of ZFC without offering a workable alternative, where "workable" means "sufficient for developing virtually all of mathematics". If you can't get Fermat's last theorem and the Poincare conjecture, then you don't have a viable alternative to ZFC.

Jim Belk
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  • Is it fair for me to say that your direct answer to the question (What good is infinity?) is "We don't really have enough information to answer the question at this time."? That seems like a perfectly reasonable response, I just want to make sure I'm not misrepresenting you. It sounds like your argument is "ZFC has worked pretty *** well. Other things we haven't tried enough to know if they work well." – Eric Stucky Dec 25 '13 at 01:40
  • The axiom of infinity is even used by finitists to argue that infinity is inconsistent!! :-) – Asaf Karagila Dec 25 '13 at 02:02
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    @EricStucky: I think this answer is much less non-commital than that. It sounds like it's a clear "pretty much everything" rather than a "we don't know". – Malice Vidrine Dec 25 '13 at 04:42
  • @MaliceVidrine: Eh, I understand what you're seeing, but I think we're talking past each other. Yes, infinity gives us a lot (in a technical sense, "almost everything"). We know that, definitely, nobody's arguing that. But what I am saying is, and what I think Jim was getting at, is that we don't know what not-infinity gets us. It could be also "almost everything". Now, I don't know what is known; perhaps we know a lot about what not-infinity gets us. But it sounds like that's what he was saying. – Eric Stucky Dec 25 '13 at 05:47
  • "What does not-infinity get us" is a rather different question than what you asked. But the answerer seems to have definite opinions on whether it's plausible to get much out of finitistic mathematics. For the record, and perhaps this is helpful to you, but what you get with not-infinity and the rest of ZF is something exactly as strong as PA. – Malice Vidrine Dec 25 '13 at 07:50
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    [If you can't get Fermat's last theorem and the Poincare conjecture, then you don't have a viable alternative to ZFC. ]

    You seem to take it for granted that ZFC is consistent. Has anybody proved it?

     – Makoto Kato Dec 25 '13 at 17:51
  • @EricStucky I agree that we don't know what throwing out the axiom of infinity would get us, and that finitary mathematics has the potential to be more powerful or provide more insight than ZFC. But I also don't find philosophical arguments in favor of finitary mathematics particularly compelling, because I don't think philosophy is a good way to resolve arguments like this. – Jim Belk Dec 25 '13 at 22:20
  • (continued) There is lots of empirical evidence that ZFC provides a good foundation for mathematics, so I would need some really compelling evidence about the effectiveness of finitary mathematics before I would be willing to switch. At this point, I'm not aware of any significant evidence that finitary mathematics is helpful, e.g. any significant new result that finitary ideas helped to prove, and as far as I know you can't even recover any meaningful portion of modern mathematics without the Axiom of Infinity. – Jim Belk Dec 25 '13 at 22:23
  • Jim, it's a joke. We can prove that $\sf ZFC$ is consistent if we allow more axioms. We can prove that theory consistent by adding even more axioms, and so on and so forth. I use that when people try to be clever and say "Ohh... but you can't prove that $\sf ZFC$ is consistent!". I don't know why, but for a moment there I thought that the two comments preceding mine were aimed at the one preceding them. Now I see that I was just tired and my mind was playing tricks on me... :-) I'll delete it, and this comment later on. – Asaf Karagila Dec 26 '13 at 16:32
  • I guess I like your answer, but I don't support to state opinions as if they were fact. In particular, the claim "[The reason is that] philosophical arguments aren't a very good method for making decisions." – Nikolaj-K Jul 16 '15 at 15:26
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Subjecting myself to these seven or so minutes of absolute horror, let me make a few comments on the video and on the ideas it presents.

  1. The historical review is skewed and out of context. It also omits the fact that Hilbert, von Neumann, Dedekind, and many others of the great mathematicians of the last century and a half have in fact accepted and welcomed the idea of infinite sets.

    The way he jumps from 1900 to 2009 is presented as if "someone brainwashed mathematicians for the last hundred years, but I have left the matrix!", and not that in fact this was a very natural progress. And that people accepted infinite sets because it is interesting.

  2. These darn "axiomatics", that nobody uses. Well, sure, but I also don't use directly single oxygen atoms. Of course the fact that I breathe dioxide molecules (amongst other things), and my body uses them for me, doesn't mean that I use them.

  3. Finally, judging by his video, his mind is limited to what he can see. Infinite sets don't really exist. Sure, we can't prove their physical existence, and not many mathematicians would claim otherwise. But is mathematics really about reality? Thank goodness, it's not. At least not in the last two centuries or so.

    And I won't even begin to tackle deep seated philosophical issues like epistemology and so on.

Now, what do infinite sets give us? First of all, they give us rich and beautiful theories where one doesn't always have to run around and count and bound everything. Exactly by not being able to count them from top to bottom, infinite sets allow us to prove very beautiful theorems on them.

The classical, and most awesome, example is indeed Ramsey theorem. The direct proof of the finite case amounts to counting arguments which are involved and annoying; the infinite proof is neat, simple and involves the compactness theorem.

This is exactly the same argument in favor of the axiom of choice, by the way, it allows us to generalize things, and prove them neatly, using the fact that certain objects exist; otherwise one has to start running around combinatorial arguments and things get messy.

But. There is no real issue with rejecting the existence of infinite sets. That's fine. But this is a philosophical belief. And much like any other belief, one has to think about it on their own and decide. You may decide to reject evolution (despite overwhelming evidence), but that means that you are probably not going to be well received in a modern academic society. Instead, you might want to consider hanging out with the good people of "Our Lord The Creator College of Creationism" instead.

The question you ought to ask yourself is this: what is mathematics? If for you mathematics is the science of counting change correctly, doing your taxes (voodoo and witchcraft), and becoming a carpenter, then by all means. There is no need for you to accept infinite sets, at all. There is also no need for you to look at any painting except blueprints, listen to any music other than your jigsaw and hammer, or enjoy any meal other than tasteless nutrients.

If for you mathematics is an amazing abstract idea, which coincidentally ends up modeling a lot of physical phenomenon, then it's probably going to be a good idea if you accept the idea of infinite sets.

Let me finish with something that I have said more than once before. I don't have a problem with finitism, or even ultrafinitism. I have an issue with how a lot of those people argue against infinitism, allow me to refer you to Louis CK, for the point which I took from him (he's talking about evolution).

Oh yeah, in the midst of writing the answer it seems that I forgot to address the issue about the continuum hypothesis.

We can assign cardinalities to each set, that's not the problem. And infinite sets do have this certain "pathology" to them, where a proper subset can have the same cardinality as the original set.

The issue with $2^{\aleph_0}=\aleph_2$ (or any other consistent value) is that the usual axioms, id est $\sf ZFC$, cannot decide the exact value.

Community
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Asaf Karagila
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  • Louis makes a very good point about people taking movies of their children, and posting on Facebook or the like, instead of actually watching at the time. Let me see if I can find that. Well, here is a sort of transcript https://www.facebook.com/louisckquotes/posts/452645034822971 I see, an embedded youtube thingie as well. – Will Jagy Dec 25 '13 at 04:54
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    Yes. I am familiar with his work. And I agree with him, that's why I don't video my kids, or post such videos on Facebook. Or have a Facebook account. Or kids. – Asaf Karagila Dec 25 '13 at 05:06
  • Alright. But what about Ziva leaving NCIS? – Will Jagy Dec 25 '13 at 05:11
  • I don't watch that. So I don't know about that. – Asaf Karagila Dec 25 '13 at 05:14
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    And I thought I knew you. – Will Jagy Dec 25 '13 at 05:15
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    +1 for "I have an issue with how a lot of those people argue against infinitism". This is what initially made me reject Wildberger out of hand; only now with a more mature mind am I able to try to sort through what he says that does and doesn't make sense. – Eric Stucky Dec 25 '13 at 07:21
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    @Eric: As I said, this is a matter of philosophical bent. I find a lot of beauty in infinite objects, and that is enough for me. – Asaf Karagila Dec 25 '13 at 07:46
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My immediate response is it that allowing infinity gives a sort of completeness. But as the person pointed out, it doesn't finish the job, it just (dramatically) kicks down the line where the new notion of "too big" is.

Maybe I'm misunderstanding the question, but I thought that's the whole point of the mathematical notion of infinity - to articulate in a workable way what "too big" means in any given case.

As others have said, if you get rid of the current framework, how are you going to prove basic things like continuity?

In a way, I can understand where Wilberger is coming from, but I think his criticism is misplaced. In my opinion, the criticism about infinity has a place in statistics. There is a lot of concern about what the distribution is when your sample (or population) is infinite, and markedly less (in my opinion) concern for small sample statistics and making appropriate inferences given just the sample that you do have. In a lot of instances you don't have the luxury of having large samples.

rocinante
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  • FYI: Wildberger poses a definition of limits in MF106 (minute 28) which leads naturally to a definition of continuity. I don't know how good a definition it is, but it seems like at least every Wildberger-continuous function is regular-continuous, which is about as good as you could expect. (Not to say I endorse this definition, just saying there is a relatively natural way to do it) – Eric Stucky Dec 25 '13 at 03:05