ThmDex – An index of mathematical definitions, results, and conjectures.

Zorn's lemma

Formulation 0

Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that

(i)	$\mathsf{Chains}(P)$ is the D4636: Set of chains in $P$
(iii)	$\max(P)$ is the D4464: Set of maximal elements in $P$
(iii)	\begin{equation} \forall \, C \in \mathsf{Chains}(P) : \exists \, M \in X : C \preceq M \end{equation}

Then \begin{equation} |\max(P)| \geq 1 \end{equation}

Formulation 1

Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that

(i)	$\mathsf{Chains}(P)$ is the D4636: Set of chains in $P$
(iii)	$\max(P)$ is the D4464: Set of maximal elements in $P$
(iii)	\begin{equation} \forall \, C \in \mathsf{Chains}(P) : \exists \, M \in X : \forall \, x \in C : x \preceq M \end{equation}

Then \begin{equation} |\max(P)| \geq 1 \end{equation}

Remarks

▶	Remark 0 (Zorn's lemma in words) If every chain in a poset has an upper bound, then the poset contains at least one maximal element.