---
tag: XGAMQBSE1G
kind: concept
name: Naive set theory
msc: ["03E30", "03A05"]
specializes: [RTHHCX84HN]
uses: [RTHHCX84HN, 0BSJ7RQ9BW, 0HESMMSYEG, 0HE5WDQ9FJ]
sources:
  - url: https://en.wikipedia.org/wiki/Naive_set_theory
same_as:
  wikidata: Q903783
  wikipedia: Naive set theory
---

## Summary

Naive set theory is [[set theory|RTHHCX84HN]] done in ordinary language, without a formal list of axioms.
In its original form, every property defines a [[set|0BSJ7RQ9BW]], the set of all objects with that property.
This principle leads to contradictions, the best known being [[Russell's paradox|0HESMMSYEG]].
The axiomatic theories, such as [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]], restrict how sets may be formed to avoid them.
The name is also used for the informal set theory taught at the start of most courses, which is consistent with those axioms.

## Explanations

### L1 human

Naive set theory treats a set as any collection you can describe.
This works well for everyday sets, such as the even numbers or the points of a circle.
But consider the set of all sets that are not members of themselves.
If it is a member of itself, then it is not, and if it is not, then it is.
So "any collection you can describe" is too generous, and the axioms of modern set theory say more carefully which sets exist.