Mathematicians who are finitists.

[A Toy Windmill]

Finitism is stricter than simply assuming there are no infinite sets. The finitist goes so far as to say that one cannot even quantify over the natural numbers! The most cited formal system which tries to capture what limited reasoning is left is "Primitive Recursive Arithmetic."

The finitist will accept that "n + 1 > n", but will regard this as a schema. to be filled in with particular numbers that they can muster.

Well, that does depend on the brand of finitism you subscribe to. Yours is one of the more extreme versions.

It's the standard one, advanced by Hilbert and maintained by Goodstein (who was mentioned in the OP). For a nice introductory paper, see Hilbert's On the Infinite. That paper doesn't give a formal system, but primitive recursive arithmetic is usually taken to scope finitist reasoning.

Finitism is still being developed, in the sense that there is still interest in just how much of contemporary mathematics could, in principle, go through in things like PRA.