PH, when you break everything down as far as you can go, you end up with

*postulates* (I realize you know what postulates are, but since I just learned them I'm going to italicize it anyway

) And postulates are merely laws defined by logic. You can define addition with sets (whatever that means, brief overview of the peano axioms and all I got was that they were laws of arithmetic formed by postulates) but then you have to define sets, and then you have to define the definitions of sets, until you can't use anything to prove stuff, and when that happens, what then? I'll tell you, Mathematicians get picky...