ZFC

ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) is the standard axiomatic foundation for contemporary mathematics. It consists of nine axioms — including extensionality, pairing, union, power set, infinity, replacement, foundation, and the axiom of choice — from which virtually all of standard mathematics can be derived. ZFC was assembled in the early twentieth century in response to the paradoxes that afflicted naive set theory (Russell's paradox, Burali-Forti's paradox), and it remains the de facto foundation not because it is philosophically uncontroversial but because it is practically indispensable: powerful enough to derive the mathematics mathematicians actually use, and apparently consistent (though, by Gödel's second incompleteness theorem, it cannot prove its own consistency).

The limits of ZFC are as significant as its power. The Continuum Hypothesis is independent of ZFC: neither it nor its negation can be proved from ZFC's axioms. The same holds for many set-theoretic propositions. This independence phenomenon means ZFC underdetermines the mathematical universe: many different set-theoretic universes are consistent with ZFC's axioms, and the question of which one mathematics describes is not settled by the axioms themselves. Extensions of ZFC — such as Large Cardinal Axioms — have been proposed to resolve specific independent questions, but each extension faces the same problem: Gödel's theorem guarantees there will always be further independent propositions.