Mathematicians who are finitists.

[polymath257]

Hmmm....there can be no actual proof of the existence of an infinite set because the collection of hereditarily finite sets is a perfectly good model of set theory without the axiom of infinity. That shows the internal consistency of the axioms. Again, in the finitist system, the *collection* of rational numbers isn't something that can be constructed. All that can be done is saying whether particular constructs are rational numbers or not.

The point is that finitistic math has very strict rules for set formation and those rules do NOT allow talking about any infinite sets.

Now, this is not to say that allowing infinite sets isn't better in many ways. It gives wonderful insights into a large variety of phenomena. It makes many arguments much easier and produces results that finitistic reasoning cannot, even results concerning finite sets.

Here's a proof by John Hopcroft, Jeffrey Ullman and Rajeev Motwani in their Introduction to Automata Theory, Languages, and Computation Edition 3 textbook, page 9:

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.