---
tags:
  - set-theory
  - theorems
---

# Axioms of set theory

The basic notions of set theory are defined in
[Basic properties of sets](Basic%20properties%20of%20sets.md). There we
introduced a formal syntax that will be utilised to define the axioms. For easy
reference:

- variables $a,b,c,...$ to range over sets
- variables $x,y,z$ to range over ordinary objects as well as sets.

## Axiom of Extensionality

Sets which contain the same members are the same set. If sets A and B contain
the same elements then A = B.
$$\forall a \forall b [\forall x (x \in a \longleftrightarrow x \in b) \rightarrow a =b]$$

[link to test](beta.md)
