---
tag: 0N109NG5ZX
kind: statement
role: theorem
name: Well-ordering theorem
aliases: [zermelo-theorem, well-ordering-principle]
msc: ["03E25"]
uses: [0F1D6PK9RZ, 0HTGB13FE2]
sources:
  - url: https://en.wikipedia.org/wiki/Well-ordering_theorem
    locator: lead section
same_as:
  wikidata: Q1457052
  wikipedia: Well-ordering theorem
---

## Statement

Assume the [[axiom of choice|0HTGB13FE2]].
Then each [[set|0SJ3RX9TXA]] $A$ has a [[well-order|0F1D6PK9RZ]].

## Remarks

Another name is Zermelo's theorem.
Over ZF, it is equivalent to the axiom of choice and to [[Zorn's lemma|04RXTQA5ZA]].
A well-order of $\mathbb{R}$ is hard or impossible to visualize, because its construction needs the axiom of choice.
With a well-order, each set allows proofs by [[transfinite induction|0ZHQMQ2C1N]].

## History

Ernst Zermelo proved the theorem in 1904 with the axiom of choice, which he formulated for this purpose.

## Explanations

### L1 human

The theorem says that each set has an order with a first element, a next element, and so on, possibly beyond infinity.
For $\mathbb{N}$ the usual order works.
For $\mathbb{R}$ such an order exists, but it is very hard to picture.