>>9372383

>You have to make absurd assumptions about the realm of ideas.

There is no transcendent, actually-existing "realm of ideas," only that which is constructed from the assumptions and definitions we make. All "mathematical truth," without exception, relies on assumptions.

Mathematics is "true" in the sense that when you lay out a set of formalized rules for abstract reasoning and symbolic manipulation, and then make a set of unambiguous definitions, what those definitions entail/imply is a priori true, as nothing more than a property of the rule system you're using.

The "realm of ideas" is not some parallel universe which we can observe, over which "sets can be infinite" is somehow a less "empirically valid" assumption than "sets can only have finitely many elements" - because here we have no robust notion at all of empirical truth/observation/evidence. The only "realm of ideas" it even makes sense to talk about is that which follows from the conceived rule sets and definitions

>>9372407

This.

Literally just " 'sets can be infinite' implies X, Y, Z; 'sets must be finite' implies X', Y', Z'." There is no need in the first place to defend any axiomatic system as "more valid."

In fact, it's nonsense to talk about validity of an axiomatic system (relative to this illusory "absolute realm of ideas") at all. The only invalid axioms are invalid through a contradiction they imply within the theory.