Definition talk:Well-Ordering

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On this page in the definition section on line 3: "has a minimal element under $\preceq$"

On the well-founded page in the definition section on line 2: "has a smallest element."

-Jshflynn 29/6/12

Good call. Replaced "Minimal" with "Smallest". In a woset, of course, the two are the same, as a woset is a toset and Minimal Element in Toset is Unique and Smallest. But it's good to be consistent.
BTW here's a technique you might want to use: when writing a post in a talk page, press the button above the edit pane which has a squiggle in it. It will add two dashes followed by four tildes. The latter will be replaced by the mediawiki s/w with your username (known as your "sig"). Like what follows here for me: --prime mover 20:56, 29 June 2012 (UTC)

recent change

Why the recent change? IMO it's useful to link to well-founded.

Is the current definition of well-founded inapplicable in this context or something? --prime mover (talk) 08:01, 5 April 2013 (UTC)

That should be fixed by importing the other definition from Definition:Well-Ordered Set, and then gutting that page. Two options for defining are:
  1. Total ordering which is a well-founded ordering (which should not just be called "well-founded", because that mostly means something else, as far as I can tell)
  2. Ordering such that every nonempty subset has a smallest element.
Note that the ordering $\{ (x,x), (y,y) \}$ on $\{ x, y \}$ is well-founded ($x$ and $y$ are both minimal), but not a well-ordering (no smallest element). --Dfeuer (talk) 13:35, 5 April 2013 (UTC)
I'm going to have to revert all the changes you are doing today because they are in danger of changing the validity of the exising definitions. As usual there are no sources quoted. --prime mover (talk) 18:13, 5 April 2013 (UTC)
Please stop, take a breath, and consult the sources. The previous definitions were wrong. Not just "non-optimal" or "not right in all theories". They were "not what anyone means by those terms". --Dfeuer (talk) 18:41, 5 April 2013 (UTC)
As for the validity of things that link there, I'm working on it. It will take a bit of time. Many of the links assume the usual meanings and are now valid when they were not before. Any exceptions will be dealt with. --Dfeuer (talk) 18:43, 5 April 2013 (UTC)
As for how this came to be, it appears that some contributor(s) got mixed up about the meanings of "minimal" and "least", arbitrarily swapping them around, and then the problem cascaded a bit. --Dfeuer (talk) 18:47, 5 April 2013 (UTC)
Okay I think I can see where it all went wrong now ... part of the original confusion between minimal and smallest elements. Okay, I'm with you now.
Still think it's worth separating out "well-founded set" and "well-founded relation" perhaps transcluding them both in a parent page "well-founded".
Okay, sorry for interrupting - but can you put "SourceReview" templates on all pages where you change stuff around so I have a flag to go and check and make sure it's all pointing in the right direction? --prime mover (talk) 18:55, 5 April 2013 (UTC)
Well-founded set isn't even on the table yet, as far as I know, but it probably will be (as a special case of well-founded relation). Well-founded ordering/ordered set and well-founded relation/relational structure (currently and I believe legitimately also called foundational) are on the table, and neither is a special case of the other. I will gladly put in some templates. --Dfeuer (talk) 19:00, 5 April 2013 (UTC)
"Well-founded set" as in "well-founded relational structure" then, whatever - in the same way you say "well-ordered set" to mean a relational structure where the relation is a well-ordering, innit? --prime mover (talk) 19:05, 5 April 2013 (UTC)

Will check sources, but I think a well-founded set/class is probably one on which the epsilon relation is well-founded. --Dfeuer (talk) 19:34, 5 April 2013 (UTC)