What good is infinity?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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).

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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.