---
tag: RTHHCX84HN
kind: concept
name: Set theory
msc: ["03E30", "03E99"]
uses: [0BSJ7RQ9BW, 0EZ758XE73, 0HE5WDQ9FJ]
sources:
  - url: https://en.wikipedia.org/wiki/Set_theory
same_as:
  wikidata: Q12482
  wikipedia: Set theory
---

## Summary

Set theory is the branch of mathematical logic that studies [[sets|0BSJ7RQ9BW]], collections of objects called their [[elements|0EZ758XE73]].
Cantor created it in the 1870s, with his study of infinite sets and their sizes.
In its axiomatic form, most often [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] with choice, it serves as the standard foundation of mathematics.
Its own subjects include ordinals and cardinals, independence results, large cardinals and descriptive set theory.

## Explanations

### L1 human

Almost every mathematical object can be built from sets.
For example, the number $2$ can be taken to be the set $\set{\emptyset, \set{\emptyset}}$, and a function is a set of ordered pairs.
Set theory writes down the basic rules for sets, and then studies what follows from them.
A famous discovery is that infinite sets come in different sizes: there are more real numbers than natural numbers.
Another is that some natural questions, such as the continuum hypothesis, cannot be settled by the usual rules.