Talk:Proper Well-Ordering determines Smallest Elements

Am I missing something here, or does the supposition that each initial segment is a set render this theorem a triviality with a very wordy and convoluted proof? --Dfeuer (talk) 20:24, 25 December 2012 (UTC)

Not sure. I don't follow your reasoning on this one. --Lord_Farin (talk) 22:03, 25 December 2012 (UTC)

I didn't give you any to follow. Basically, this theorem should be refactored into several, each simple. Something like this:

• Each well-ordered class has a minimal element (the special case of $S=B$).
• Each subclass of a well-ordered class has a minimal element.
• Minimality in subclass with respect to an endorelation is equivalent to minimality under the inducted endorelation(trivial, methinks).
• $x$ is minimal in $B$ iff $x \in B$ and $S_x \cap B = \varnothing$. --Dfeuer (talk) 23:11, 25 December 2012 (UTC)

First point appears unvalid since $S$ may not be a subset of $S$. Similar for second point. Third and fourth point seem to be accurate. --Lord_Farin (talk) 13:21, 26 December 2012 (UTC)

I'm pretty sure the "subset" is an error in the current version. What do you think? --Dfeuer (talk) 13:57, 26 December 2012 (UTC)

All I will (continue to) say is that this part of the site is currently practically useless in its idiosyncrasy and imprecision. I don't know, since I never studied class theory in any meaningful depth; I stopped when I got its idea. Grothendieck's solution by means of set universes is of greater appeal to me. --Lord_Farin (talk) 14:01, 26 December 2012 (UTC)