Category:Zorn's Lemma

This category contains pages concerning Zorn's Lemma:

Formulation 1

Let $\struct {S, \preceq}, S \ne \O$ be a non-empty ordered set such that every non-empty chain in $S$ has an upper bound in $S$.

Then $S$ has at least one maximal element.

Formulation 2

Let $\struct {S, \preceq}, S \ne \O$ be a non-empty ordered set.

Let $T \subseteq \powerset S$ be the set of subsets of $S$ that are totally ordered by $\preceq$.

Then every element of $T$ is a subset of a maximal element of $T$ under the subset relation.