Okay. I have several sources of skepticism about infinite sets. One has to do with my never having observed a large cardinal. One has to do with the inability of first-order logic to discriminate different sizes of infinite set (any countably infinite set of first-order statements that has an infinite model has a countably infinite model - i.e. a first-order theory of e.g. the real numbers has countable models as well as the canonical uncountable model) and that higher-order logic proves exactly what a many-sorted first-order logic proves, no more and no less. One has to do with the breakdown of many finite operations, such as size comparison, in a way that e.g. prevents me from comparing two "infinite" collections of observers to determine anthropic probabilities.

The chief argument against my skepticism has to do with the apparent physical existence of minimal closures and continuous quantities, two things that cannot be defined in first-order logic but that would, apparently, if you take higher-order logic at face value, suffice respectively to specify the existence of a unique infinite collection of natural numbers and a unique infinite collection of points on a line.

Another point against my skepticism is that first-order set theory proper and not just first-order Peano Arithmetic is useful to prove e.g. the totalness of the Goodstein function, but while a convenient proof uses infinite ordinals, it's not clear that you couldn't build an AI that got by just as well on computable functions without having to think about infinite sets.

My position can be summed up as follows: I suspect that an AI does not have to reason about large infinities, or possibly any infinities at all, in order to deal with reality.