Are there any other rigorous or serious attempts at understanding numbers besides set theory?
There is Peano Arithmetic, a first-order theory that axiomatizes natural numbers. It does not rely on axioms of set theory. Also, one can consider a category or a topos with a natural number object. From Wikipedia:A traditional axiomatic foundation of mathematics is set theory, in which all mathematical objects are ultimately represented by sets (even functions which map between sets). More recent work in category theory allows this foundation to be generalized using toposes.