> These are generally called axioms, but these are totally distinct from axioms in the sense of ZFC's axioms. I wouldn't really say they're "totally distinct".

> These are generally called axioms, but these are totally distinct from axioms in the sense of ZFC's axioms. I wouldn't really say they're "totally distinct". For example, one could say that general group theory is studying the theory (in the logic sense) of a group (usually the second order theory rather than first order, but you get idea). Then adding assumptions about the structure of the group under consideration is like adding axioms and looking at the new theory.