>>13749495I think there's a distinction to be drawn here about what axioms are. Formal systems start with certain kinds of symbols (a, b, c,... 1,2,3,..., +,-,=,...) plus rules for combining them (1+2 can be replaced by 2+1; T AND F can be replaced by F and T OR F by T; two lines in a projective plane intersect at a unique point; etc).
Godel's argument essentially provides you a procedure, which you can apply to any sufficiently-complicated formal system, by which you arrive at a contradictory result -- it's something along the lines of generating "this sentence is false" in the given formal system. It doesn't depend on the details of the system in question, just that enough rules are present to follow the same argument Godel did. (Specifically, that we can add and multiply numbers -- or do something sufficiently analogous, like the set theory constructions.)
If we do that, then the sentence we've made will either be unprovable or self-contradictory (or else the system isn't powerful enough, but then we can't usually prove interesting results). This applies to ZFC, the standard axiomatic foundation of math, as well as various other formulations (eg a slightly more formal axiomatization of Euclid's Elements).
