---
tag: JRBRNZT680
kind: definition
name: Set theory
msc: ["03E30", "03E99"]
defines: RTHHCX84HN
uses: [0SJ3RX9TXA, 0EZ758XE73, 086QKPF0A6, 0HE5WDQ9FJ, 0TB16T2C97, 0MN2MER9WN]
sources:
  - url: https://en.wikipedia.org/wiki/Set_theory
---

## Definition

*Set theory* is the branch of mathematical logic whose objects of study are [[sets|0SJ3RX9TXA]] and the membership relation $\in$ between an [[element|0EZ758XE73]] and a set.
An *axiomatic set theory* is a theory in [[first-order logic|086QKPF0A6]] with axioms about $\in$.
Its language has the binary relation symbol $\in$, and possibly further symbols.
The standard axiomatic set theory is ZFC, [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] with the axiom of choice.

## Remarks

Example: ZFC is an axiomatic set theory.
Its language has the single relation symbol $\in$, and its axioms, such as the [[axiom of extensionality|0TB16T2C97]], speak only about $\in$ and equality.
Non-example: Peano arithmetic is not a set theory in this sense, because its language has $0$, successor, addition and multiplication, and its objects are natural numbers.
Other axiomatic set theories include Zermelo set theory, von Neumann–Bernays–Gödel set theory, Morse–Kelley set theory and Quine's New Foundations.
Questions such as the [[continuum hypothesis|0MN2MER9WN]] belong to set theory and are independent of ZFC, if ZFC is consistent.