In that case, I would say you're a finitist (which, as the name seems to suggest, is fine). The set theory approach gives seemingly clear answers to all of the

In that case, I would say you're a finitist (which, as the name seems to suggest, is fine). The set theory approach gives seemingly clear answers to all of these questions by talking about domains and ranges of functions, rather than talking about what is or isn't an "inexhaustible number". There are potentially different correspondences available, represented by different functions; some possible correspondences may follow the "if you select corresponding to itself in preference" pattern, while others don't. = and ≤ for cardinalities are defined using existence of certain functions between sets. However, you don't have to believe that any infinite sets exist or that we should be allowed to quantify over them, or that we should attempt to define cardinality for infinite sets at all. Still, as Dana Scott said in a related context, "if you want more, you have to assume more".