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| | <blockquote style="display:table;background:#eeeeee;padding:10px;" class="toccolours"> | | <blockquote style="display:table;background:#eeeeee;padding:10px;" class="toccolours"> |
| − | ==[[Zorn's Lemma]]==
| + | Inactive. |
| − | '''Zorn's Lemma''' is a [[set theory | set theoretic]] result which is equivalent to the [[Axiom of Choice]].
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| − | Let <math>A</math> be a [[partially ordered set]].
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| − | We say that <math>A</math> is ''inductively ordered'' if every [[totally ordered set | totally ordered]] [[subset]] <math>T</math> of <math>A</math> has an upper bound, i.e., an element <math>a \in A</math> such that for all <math>x\in T</math>, <math>x \le a</math>. We say that <math>A</math> is ''strictly inductively ordered'' if every totally ordered subset <math>T</math> of <math>A</math> has a [[least upper bound]], i.e., an upper bound <math>a</math> so that if <math>b</math> is an upper bound of <math>T</math>, then <math>a \le b</math>.
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| − | An element <math>m \in A</math> is [[maximal element | maximal]] if the relation <math>a \ge m</math> implies <math>a=m</math>. (Note that a set may have several maximal... [[Zorn's Lemma|[more]]]
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| | </blockquote> | | </blockquote> |
