Mathematicians who are finitists.

[polymath257]

Sorry, but it doesn't prove what you claimed. This proposition says that if S is finite and U is infinite, then T=U-S,
the complement of S in U, is infinite.

This does NOT prove the existence of an infinite set. It shows that *if* there is an inifnite set U, then there are other
infinite sets (U-S where S is finite).

Now, show how to get that set U from finitist principles.

Proofs that there are infinitely many primes

And the finitist would not accept that there is a set of primes. Instead, they would say that for every finite set of primes, there is a prime not in that set.

For example, the orignal proof by Euclid was of this form. And  by modern standards, Euclid would have been a finitist. The topological proof would certainly NOT be accepted by finitists. Some formulation of the Goldbach proof would probably be acceptable (again, for any finite set of primes, there is a prime not in the list). Kummer *was* a finitist.

Once again, for a finitist, NO infinite collection is acceptable: only finite collections.

It is possible to talk about the *property* of being a natural number, but not the *set* of all such numbers. It is possible to talk about the *property* of primality, but not the set of all such. So, a finitist would simply say there is no 'collection of all primes'.

But this is similar to a modern set theorist saying there is no 'collection of all sets' since it is provably the case that this collection is not a set.