Definition talk:Well-Founded Ordered Set

Is this the same as Definition:Well-Ordered Set, or is there some subtle difference I'm missing? --Cynic (talk) 20:21, 8 June 2009 (UTC)

Subtle difference. A well-ordered set is totally ordered, a well-founded set may not be, as it's defined on a partially ordered set. I'm about to post up a few pages on this subject on the way towards some work on the ordinals and cardinals. Infinity will never seem so simple again ... --Prime.mover 20:32, 8 June 2009 (UTC)

Different definitions with different strengths
I think we really want all of these, in some form.

From weakest to strongest (approximately):

A relational structure is well-founded iff:
 * 1) It has no infinite descending chains.
 * 2) Every non-empty subset has a minimal element. (Takeuti)
 * 3) Every non-empty subclass has a minimal element. (S &amp; F)
 * 4) Well-founded induction works in the system.

The ordering is a bit approximate because different formulations of well-founded induction will likely vary a bit in strength, all stronger than 3 but not all stronger than 4. --Dfeuer (talk) 17:47, 5 April 2013 (UTC)


 * Build a parent page for now with subpages with numbers (i.e. definition 1, definition 2, etc.) and we can work out what to call them later. --prime mover (talk) 18:56, 5 April 2013 (UTC)