Dear philosophers, I really appreciate your website, which I just discovered! I'd like to make one comment regarding the recent questions about infinite sets on March 7 and March 14. In your responses (Allen Stairs and Richard Heck on March 14), you write that you do not know of any professional mathematicians who deny the existence of infinite sets. However, such mathematicians do indeed exist (although marginally). They are sometimes referred to as "ultrafinitists". One well-known living proponent of this view is Princeton mathematician Edward Nelson, also see http://en.wikipedia.org/wiki/Edward_Nelson and http://en.wikipedia.org/wiki/Ultrafinitism Specifically, one argument an ultrafinitist might use is that formal proofs are finite. Thus, although we might use the concept of infinite sets in our reasoning, there is no need to assume that infinite sets actually exist, because any mathematical statement could be preceded by the phrase "There is a finite proof of the statement that ..." I hope this makes for an interesting addition to your answers. Best regards, Sam PhD student, mathematical logic RU Nijmegen, the Netherlands

What I said was:

"It's important to distinguish two different issues: (i) whether there are infinitely many natural numbers; (ii) whether there are mathematical objects that are themselves infinite. And it is possible to accept that there are infinitely many natural numbers without accepting that there is a set of all of them or, more generally, that there are any objects that are, in their own right, infinite. And there are respected mathematicians who hold this kind of view, though they are definitely a minority."

So I was not saying that no professional mathematicians would deny the existence of infinite sets. Indeed, Nelson was very much the sort of person I had in mind. He may well be an example of a mathematician who does not think that there is a largest natural number, but who does not think that there are any infinite sets. But I'm not absolutely sure about this, due to the fact that much talk of infinite sets can be coded in the sorts of weak theories that Nelson would accept.

I think there is an interesting question about what makes a mathematical object "essentially infinite", if I can put it that way. One might think the set of natural numbers, for example, though infinite in one sense, is also finitely describable and so, in some other sense, not "essentially" infinite. One might put algebraic reals in the same class, as opposed (say) to non-recursive reals. I don't know if this notion can really be made coherent. Perhaps various forms of predicativism (e.g., Feferman's) might be thought to elaborate this kind of idea.

I don't think the mere fact that we can append "There is a finite proof that..." to any mathematical statement shows very much, because "There is a finite proof that S" and S are not equivalent, in any obvious sense. Right to left, there are Goedelian problems; left to right, one needs to assume, most intuitively, that all the axioms used in the proof are true (though, formally, some kind of reflection principle would also work).

Read another response by Richard Heck