Zorn's Lemma/Warning

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Theorem

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.


Warning

The statement of Zorn's Lemma supposes the existence of an upper bound in $S$ for any (non-empty) chain $A$.

It does not guarantee the existence of an upper bound for $A$ in $A$ itself.

The conclusion is that:

$\forall a \in S: a \le x \implies a = x$