You're absolutely right. ZFC + ¬Con(ZFC) is a weird set of axioms, and no mathematician really studies it. I skipped this point, because we weren't really discu

You're absolutely right. ZFC + ¬Con(ZFC) is a weird set of axioms, and no mathematician really studies it. I skipped this point, because we weren't really discussing such axioms, but it's an important point that there are different levels of mathematical / philosophical commitments, and realism and PA are way stronger commitments than the ones people have about cardinalities.