# ZFC Axioms

**Type:** Concept  
**Domain:** Foundations & Logic  
**Codex URL:** /mathematics/foundations-logic/zfc-axioms/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Nine axioms of Zermelo-Fraenkel set theory with Choice — the standard foundation of modern mathematics.

## The nine axioms
1. Extensionality — sets equal iff same elements
2. Regularity — no set contains itself
3. Specification — subsets via well-defined properties
4. Pairing — sets of two elements
5. Union — combine collections of sets
6. Replacement — functions on sets yield sets
7. Infinity — an infinite set exists
8. Power Set — all subsets form a set
9. Choice — simultaneous selection from any collection of non-empty sets

## Why needed
Russell's Paradox (1901) — naive unrestricted set theory leads to contradiction. ZFC (Zermelo 1908, Fraenkel 1922) carefully restricts constructions to avoid this.

## Axiom of Choice controversy
Existence claim without explicit construction method. Leads to Banach-Tarski Paradox. ZF (without C) vs ZFC (with C) — the "C" makes it the standard default.

## Cannot prove its own consistency
Gödel's Second Incompleteness Theorem — ZFC cannot prove itself free of contradiction using only its own resources.

## Independence results
Continuum Hypothesis — neither provable nor disprovable from ZFC (Gödel 1940 + Cohen 1963).

## Sources
### Tier 1
- Zermelo, E. (1908). Mathematische Annalen.
- Jech, T. (2003). Set Theory. 3rd millennium ed.
### Tier 2
- Enderton, H.B. (1977). Elements of Set Theory.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*