ZFC
Zermelo–Fraenkel set theory with the Axiom of Choice — a standard foundation for modern mathematics
ZFC is Zermelo–Fraenkel set theory together with the Axiom of Choice — the usual axiomatic foundation in which most contemporary mathematics (analysis, algebra, topology, and much else) is formalised. Mathematicians often say a theorem is “provable in ZFC” when it follows from these axioms.
ZFC is not the same as Peano Arithmetic (PA). PA axiomatises arithmetic directly; ZFC builds numbers (and far more) from sets. PA can be interpreted inside ZFC, but the systems serve different roles in practice.
How ZFC fits the epistemic ladder (same pattern as PA):
- Tier 0 (Rⁿ): mathematical structure — not identical to the axiom list.
- Tier 1: formal captures — stated axioms, proof steps, adopted foundations committed in a log.
- Tier 2+: claims such as “theorem of ZFC” — necessary relative to tier-one axioms and proofs, still fallible as captures of Rⁿ.
Like PA, ZFC has meta-mathematical limits (Gödel, independence results such as the continuum hypothesis relative to ZFC). Those limits are about what the formal system can settle, not a licence to treat tier-one records as tier zero.