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 ... --Matt Westwood 20:32, 8 June 2009 (UTC)

Well-Founded vs. Foundational
Are these terms usually used for different things as they are here? Some source review is probably in order. --Dfeuer (talk) 17:36, 11 March 2013 (UTC)


 * Usually a good idea, having raised an issue, to wait for a response before going ahead with changes. --prime mover (talk) 08:02, 5 April 2013 (UTC)


 * ... having said that, it *has* been a few weeks since you raised it. --prime mover (talk) 10:26, 5 April 2013 (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)


 * Different page and different name for each page, all complete with sources, and a page including the difference between all of them and which imply which, and which ones are used to define which theorems. At the moment what we have (if you scrub Takeuti) hangs together. If you go and change existing definitions it might not. Therefore unless you take the approach I have specified, it will be consistently reverted. --prime mover (talk) 18:18, 5 April 2013 (UTC)


 * I can add #3 with source today (from S &amp F). The only problem is naming. Exactly the same problem occurs wrt well-orderings. --Dfeuer (talk) 18:54, 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)