Mathematicians who are finitists.

[polymath257]

Again, for a finitist, all sets are finite. So the issue simply doesn't arise. The definition of cardinality can be made, but it trivializes. No set is uncountable, no set is even countably infinite.

For a finitist, there *is* no set of natural numbers. There *is* no set of real numbers. ONLY finite sets exist in the finitist system.

I think you can see why most mathematicians see finitism as limiting.

I found Cantor's diagonalization proofs, as described by Rosen, to be on par with the Pythagorean theorem.  Seems like the finistists are engaging in ad hoc reasoning.  But, maybe they feel that there are really square circles?

I have one book (The Foundations of Mathematics by Kunen) that discusses the Philosophical differences between Platonists, Finitists, and Formalists as being similar to those who are theists, atheists, and agnostics. The Platonists believe in the existence of mathematical objects in some real sense. Finitists limit themselves to only finite sets. And formalists organize their lives so that the question isn't particularly relevant.

So, for the Platonist, the Continuum Hypothesis is either true or false. Our axiom system may not be strong enough to answer the question of which, but for the Platonist there is a definite answer.

For the finitist, the Continuum Hypothesis is simply meaningless. It talks about the sizes of different infinite sets and, for the finitist, this is literally meaningless.

The formalist is happy with any axiom system that gives nice results. The Continuum Hypothesis has no answer beside whatever axioms we *choose* that might resolve it.

For the Platonist, the set of real numbers is really a larger cardinality than that of the natural numbers. For the formalist, this statement can be proved in some systems and not in others. And for the finitist, it is just meaningless.