Talk:Zorn's Lemma Implies Zermelo's Well-Ordering Theorem

This proof seems a bit confused. (considering $X \cup \set {x_0}$ when $x_0 \in X$, awkward notation for defining $\preceq'$, conclusion doesn't make sense to me) It's not quite up to standard yet, but I've cleaned up the confusion. To do:


 * Prove that the union over a chain is an upper bound for that chain, pretty easy but should be spelt out
 * Show that $\preceq'$ is a (total) order
 * Show that $\preceq'$ is a well-order (if $A \subseteq E \cup \set {x_0}$ is non-empty, then its $\preceq'$-least element is $x_0$ if $x_0 \in A$, or its $\preceq$-least element otherwise)

Caliburn (talk) 12:32, 15 March 2022 (UTC)