---
tag: 0HE5WDQ9FJ
kind: concept
name: Zermelo–Fraenkel set theory
aliases: [zf, zfc, zermelo-fraenkel]
msc: ["03E30"]
uses: [0SJ3RX9TXA, 086QKPF0A6, 0TB16T2C97]
sources:
  - url: https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory
    locator: lead section
same_as:
  wikidata: Q191849
  wikipedia: Zermelo–Fraenkel set theory
---

## Summary

Zermelo–Fraenkel set theory, ZF, is an axiomatic theory of [[sets|0SJ3RX9TXA]] in [[first-order logic|086QKPF0A6]] with one binary relation $\in$.
Its axioms are extensionality, regularity, separation, pairing, union, replacement, infinity and power set. Separation and replacement are axiom schemas.
ZF with the [[axiom of choice|0HTGB13FE2]] is ZFC, the standard foundation of most of mathematics.

## Remarks

The [[axiom of extensionality|0TB16T2C97]] says that a set is determined by its elements.
Separation forms only subsets $\{x \in A \mid P(x)\}$ of a given set $A$. This avoids [[Russell's paradox|0HESMMSYEG]].
The [[continuum hypothesis|0MN2MER9WN]] is independent of ZFC, if ZFC is consistent.

## Explanations

### L1 human

ZF is a short list of rules that says which sets exist.
For example, for two sets there is a set that contains both, and for each set there is the set of its subsets.
From these rules, one can build the numbers, functions and most other objects of mathematics.