Zorn's lemma vs Least Upper Bound axiom - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-20T01:07:10Z http://mathoverflow.net/feeds/question/89614 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89614/zorns-lemma-vs-least-upper-bound-axiom Zorn's lemma vs Least Upper Bound axiom Thang Tran 2012-02-26T23:05:57Z 2012-02-26T23:10:55Z <p>I am confused by the Zonrn's lemma and Least Upper Bound axiom:</p> <p>(1) Least upper bound axiom: every subset of real number if has an upper bound then has a least upper bound.</p> <p>(2) Zorn's lemma: Let (A, &lt;=) be a partially ordered set. If every chain in A has an upper bound then A has maximal.</p> <p>I think if each chain in A has an upper bound then the chain should have maximal (as the Least upper bound axiom state above) hence the set of maximal of A should be the set of maximal of chains of A.</p> <p>My intuition want to believe (though I know it is wrong) that Zorn's lemma should be merged with Least Upper bound axiom into the form:</p> <p>Let (A, &lt;=) be a partially ordered set. If a chain in A has an upper bound then it has maximal and hence A has maximal.</p> <p>Could you give me an encounter-example to show that if a chain has an upper bound then it probably has no maximal.</p> <p>Thank you.</p> http://mathoverflow.net/questions/89614/zorns-lemma-vs-least-upper-bound-axiom/89616#89616 Answer by Asaf Karagila for Zorn's lemma vs Least Upper Bound axiom Asaf Karagila 2012-02-26T23:10:55Z 2012-02-26T23:10:55Z <p>Consider the set \$(0,1)\$ in \$\mathbb R\$. It has a <em>least</em> upper bound, but no maximal element.</p>