A couple of observations. It's odd that the article doesn't spell out Zermelo–Fraenkel, and instead uses the shorthand ZFC axioms. At a fundamental level, axio

A couple of observations. It's odd that the article doesn't spell out Zermelo–Fraenkel, and instead uses the shorthand ZFC axioms. At a fundamental level, axioms are simultaneously solid and slippery. The entire purpose of a set of axioms is to serve as a backstop. Without axioms there are no proofs, and instead you wind up riding infinite regress; turtles all the way down. In short, proofs without axioms are impossible. But once you have a set of axioms, proofs are precisely that, proofs. The notion that mathematics is a social compact is aggrandizing the role of the axiom, and minimizing the rest of mathematics. It is certainly the case that axioms are in fact a "social compact" in the sense that if we cannot agree that a set of given axioms are true without proof, then we can prove nothing. But the goal of axioms is to make that compact as narrow and clear as possible. The brief discussion of Godel's incompleteness theorem elides the fundamental nuance that incompleteness has only been demonstrated in the context of self reference. There is a longer and better discussion of how this relates to logic, science, and consciousness in Hofstadter's seminal work "Godel, Escher, Bach"