equivalence of Kuratowski’s lemma and Zorn’s lemma
In this entry, we prove the equivalence of Kuratowski’s lemma and Zorn’s lemma, thereby establishing the equivalence of Kuratowski’s lemma and the axiom of choice. The proof, of course, only uses ZF axioms.
Kuratowski’s lemma implies Zorn’s lemma.
Suppose is a poset such that every chain has an upper bound. Let be a chain in . By Kuratowski’s lemma, can be extended to a maximal chain , which, by assumption, has an upper bound . Suppose for some , . If , then , or , which means is a chain extending , thus extending . But this means that is not maximal, a contradiction. This shows that , or that is a maximal element in , proving Zorn’s lemma. ∎
Zorn’s lemma implies Kuratowski’s lemma.
Suppose is a poset and a chain in . We assume that . Let be the set of all chains in extending . Partially order by inclusion so that is a poset. Let be a chain in . Let . We want to prove that is an upper bound of in .
is a chain in . If , then and for some . Since is a chain in , or , which implies that belong to the same chain (either or ) in . So or . This shows that is a chain in .
is in . If , then for every , and therefore , showing that extends , or .
is an upper bound of . Pick any , for an arbitrary . Then , so . Since is arbitrary, is an upper bound of .
By Zorn’s lemma, has a maximal element . Then is a maximal chain in extending , for if there is a chain in such that , then and would no longer be maximal. ∎
|Title||equivalence of Kuratowski’s lemma and Zorn’s lemma|
|Date of creation||2013-03-22 18:44:23|
|Last modified on||2013-03-22 18:44:23|
|Last modified by||CWoo (3771)|