Mathematicians who are finitists.

[polymath257]

Well, strict finitists only have finite sets, so the question of the sizes of infinite sets simply doesn't arise. So, individual numbers can be considered, any any finite set of numbers, but the collection of all natural numbers is rejected. Since rational numbers are essentially reduced pairs of natural numbers (think fractions), a finitist can also talk about natural numbers. But it becomes much harder to even talk about real numbers, let alone the collection of *all* real numbers. So the diagonalization argument, as usually seen, doesn't abide by finitist principles.

They will admit the possibility of adding a new element to any already existing set, but not to allow the union over all such processes to get an actual infinite set.

It is possible to formulate Turing machines in such a way that the tape is only *potentially* infinite as opposed to actually infinite. The idea is that a new cell is added at either end if required. In this way, the collection of cells is finite at every step in time. This is how a typical finitist would speak of Turing machines. And, again, this is their bread and butter--finite state machines, recursive functions, etc.

The fact that the set of rational numbers are countable and the set of real numbers is uncountable is proof that actual infinities, at least in the abstract sense, are completely coherent mathematically.  Cantor demonstrated this beyond any shadow of doubting reality.  I am no expert, but have read Rosen multiple times over the years, now in its 8th edition, having first taken Discrete Mathematics with his 1st edition back in 1990.  Aho, Uullman and Hopcroft also have a proof of the existence of infinite sets that I will try to post here.

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.