# Zorn's Lemma　`lem.zorn`
English entry · root `../`
[in graph](../#n=lem.zorn)　·　[Chinese page (authoritative)](./lem.zorn.md)

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

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

## Abstract
Let $(P, \preceq)$ be a non-empty poset in which every chain has an upper bound in $P$. Then $P$ has a maximal element. Both hypotheses are needed: the empty poset has none, and an upper bound for every chain yields maximality only, never a greatest element. It is the standard engine of existence proofs — bases, maximal ideals, compactness, Hahn–Banach — and it is non-constructive: an element is produced without telling you which one.

> 续见 nodes/lem.zorn.en.2.md