Talk:Well-founded relation

message: Having written this, I now think it should be merged somehow with the page about well-founded sets.  The two pages complement each other well; there isn't much overlap, and I think both pages are good,  but I think it might be better if all the information were in one place.  Dominus 08:05, 10 Aug 2003 (UTC)
replies: text: The merge has been completed yesterday, however the text could use some editing to remove duplication and tidy up the prose. -- Ap 11:26, 17 Sep 2003 (UTC)
thread_title: Merge discussion

message: Perhaps the article should be more careful to distinguish the case of proper classes, and the importance of set-like-ness should be stressed (without this requirement we could define truth by transfinite induction, contrary to Tarski!) --Quux0r 08:17, 16 April 2007 (UTC)
replies: text: I know this I'm responding a bit late, but 1. I don't understand the remark about defining truth. 2. The requirement that the relation be set-like does not appear to be necessary if the axiom of foundation holds. --Dfeuer (talk) 18:13, 15 April 2013 (UTC)
thread_title: Set-like relations and proper classes

message: I'm far from an expert, but a small sample suggests substantial variation between different definitions. Leaving aside the descending chain condition (which I don't think is commonly used as a definition these days), there are still quite a few choices to be made:
1. Definition in terms of existence of a minimal/initial element vs. definition in terms of the validity of induction. These are classically equivalent in a rather trivial way, but not intuitionistically so.
2. Requiring the relation to be set-like (proper) or not. If so, then the next matter becomes irrelevant but generality is compromised. At least some sources separate these matters, speaking of well-founded proper relations.
3. Requiring every subclass to have a minimal element or only every subset. In the presence of the axioms of infinity, replacement, and foundation, it doesn't matter. But unless the stronger condition is met, one way or another, induction on proper classes will not work.
When it comes to the question of whether ∈ is well-founded on a class, matters are somewhat simplified, as the stronger form of (3) then follows from the weaker form given the axiom of infinity and the axiom of replacement.
Speculation: my guess is that some shy away from the strong form of (3) because it cannot be expressed directly in a first-order theory, as it quantifies over general classes. --Dfeuer (talk) 21:29, 10 April 2013 (UTC)
thread_title:  Variations in definitions 

message: For those of us not as conversant with all the terms and symbols of mathematics, a simple example near the top can greatly speed up comprehension.  Here is some possible text, which might be placed just before: "(Some authors include ..."  Please feel free to improve upon this.
For example, the relation "is less than" is wellfounded on the class of positive integers because any subset (eg, {5,3,9}) has a minimum (ie, 3).  In the formalism: none of these are true:  (5<3), (3<3), nor (9<3) -- or more generally, (s,m) does not observe the relation "is less than" for any s in the subset.
Just trying to make this accessible to a broader audience.  EJR (talk) 06:41, 10 November 2013 (UTC)
thread_title:  A simple example to help non-mathematicians? 

message: The relation R and the concept of "minimal" might be understood easier if we use the letter > instead of R. Then again, maybe it's better to use the ambiguous R which doesn't bring with it all the assumptions of the more familiar ordering on the number line.
Crasshopper (talk) 15:53, 19 January 2014 (UTC)
thread_title:  R or > ? 

message: There seems to be a discrepancy in the definition. A wellfounded relation is defined in terms of minimal elements, but they on turn are only defined (in this wiki) for partial orderings. Madyno (talk) 18:16, 5 July 2015 (UTC)
thread_title:  Definition 

message: Should there be an assumption that P(x) is true for every x∈X such that (z,x)∉R for every z∈X in order for the induction proof to work properly? Lapasotka (talk) 14:53, 18 October 2018 (UTC)
replies: text: For such an x, the property   "P(y) is true for all y such that yRx"   is trivially true, so P(x) is required to be true by the induction rule given in the article. No change of text is necessary. - Jochen Burghardt (talk) 18:47, 18 October 2018 (UTC)
replies: text: That's true.  But at the same time, this is a hard enough problem for novices to work out that it may be worth saying something about it.  I think explanatory footnotes are good for this sort of thing — they're visible without following a link, but they don't interrupt the flow as much as an inline explanation. --Trovatore (talk) 20:25, 18 October 2018 (UTC)
thread_title:  Induction and recursion 

message: The bit on induction is missing the base case. The statement as written is false. To convince yourself, take  the false statement and  any non-empty set. 2A00:1398:4:A0F:8877:8440:3C36:92D5 (talk) 11:25, 25 May 2023 (UTC)
replies: text: There is nothing wrong. The base cases (in general, there are many) are provided by the minimal elements. Indeed, if x is minimal then P(y) is true for all y such that y R x, even if P is the false statement. So, the proof of  includes the proof of P(x) for every mimimal element  x.D.Lazard (talk) 13:55, 25 May 2023 (UTC)
thread_title:  Induction Base