Mathematicians who are finitists.

[robvalue]

I find the whole thing to be semantics. If we're dealing with reality, then of course we can have a potentially unbounded and growing value, which is always finite but "potentially infinite". However, in a purely abstract mathematical setting, it’s easy to create infinite sets using particular rules. One can certainly analyse whether the results are internally consistent or not according to those rules, but the set of elements is infinite regardless.

It can be proved that the size of the set is larger than any number, and so it is infinite. The set doesn’t have to "grow" to produce all the elements, they are simply defined into abstract "existence". I suppose you could get into semantics about how exactly the elements are defined, and if that’s done inductively, then you could say the set "never finishes". Well, of course, but this is imposing an artificial limit on a process which doesn’t involve time passing in the first place. You have defined every element in the set, and there are an infinite number of them.

It is fascinating about relative infinities, as has been mentioned with the real and natural (or rational) numbers. I remember being told at uni about how if you compare the density of all the real and rational numbers in any particular interval (say between 0 and 1), the real numbers actually take up all the space. They are infinitely more numerous (thus a higher cardinality). There is no way of placing the real numbers in a list so they can be counted. You will always be missing elements between any list entrants.