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Zorn's Lemma

Zorn's Lemma is a set theoretic result which is equivalent to the Axiom of Choice.
Let $A$ be a partially ordered set.
We say that $A$ is inductively ordered if every totally ordered subset $T$ of $A$ has an upper bound, i.e., an element $a \in A$ such that for all $x\in T$ , $x \le a$ . We say that $A$ is strictly inductively ordered if every totally ordered subset $T$ of $A$ has a least upper bound, i.e., an upper bound $a$ so that if $b$ is an upper bound of $T$ , then $a \le b$ .
An element $m \in A$ is maximal if the relation $a \ge m$ implies $a=m$ . (Note that a set may have several maximal... [more]

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Zorn's Lemma