# 良序定理　`thm.wellordering`
English entry · root `../`
[in graph](../#n=thm.wellordering)　·　[Chinese page (authoritative)](./thm.wellordering.md)

> English: **draft** · machine-drafted, pending author review; the Chinese page is authoritative.

layer 9 · theorem · cluster 选择原理 (`order.choice`) · Set Theory / Order Theory

## Abstract
Every set can be well ordered: $\forall X\, \exists \preceq$ a well-ordering of $X$, equivalently every set is equinumerous with an ordinal. Zermelo introduced the axiom of choice in 1904 precisely to prove this, and over ZF the statement is equivalent to AC. Immediate consequence: any two cardinals are comparable, and $\mathbb{R}$ carries a well-ordering that cannot be written down explicitly.

> 续见 nodes/thm.wellordering.en.2.md