Today I Learned

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

Category Archives: set theory

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.