# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 167: Zorn’s Lemma

In set theory, Zorn’s Lemma states that if $S$ is a nonempty, partially-ordered set such that every chain (totally ordered subset) has an upper bound, then $S$ has at least one maximal element $m$. (That is, an element such that $m \leq n$ is not true for any $n \in S$.

Assuming the axioms of Zermelo-Fraenkel set theory, Zorn’s lemma is equivalent to the axiom of choice and the well-ordering theorem, respectively.