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I made some edits at well-order. I am removing a query box, having duly extracted some punchlines. These edits also forced an edit to partial function, where I added the generalization to partial maps in any category with pullbacks.
+– {: .query} This need not exist; in particular, $S_a$ may be empty. What do we really want to say here? (We could talk about the successor of a well-ordered set.) —Toby Mike: Yeah, or we could say that successor is a partial function. One definition of a limit ordinal is one on which successor is totally defined. =–
Good; I added redirects such as partial map.
At well-order I found it non-trivial to extract the usual classical definition. (Before I reached the second-but-last bullet item that states it, I was following a dozen links to other kinds of orderings and their properties).
For the sake of the reader, I have now given the usual definiton the first paragraph in the Definition section (I didn’t remove it from that bullet list).
But I think some expert might still make that entry a bit user-friendly.
I linked well-order and well-quasi-order. I do wonder whether there are constructions that assume an index set needs to be well-ordered that could use the weaker notion. I don’t know the consistency of asking that every set admits a well-quasi-order, but surely it’s weaker than full AC.
There is a subtlety that I believe to be wrong.
The page reads that “a well-order is precisely a well-founded total order”. The way a total order relation is defined in nLab, equality is allowed, and by applying the definition of well-founded relation on say the natural numbers, the empty set would be an inductive subset. So what it should read is that
$\text{ A well-order is precisely $\lneq$ for a well-founded total order $\leq$ }$Or one could just add the notion of a strict order (orders and strict orders are in one to one correspondence), and just say a well-founded strict total order.
Sorry I screwed up the previous comment, (EDIT: and I can’t change it because i wasn’t logged in), I meant to write
$\text{A well-order is precisely } \lneq \text{ for a well-founded total order } \leq$Amazingly, here you can edit your comments indefinitely. Just hit “edit” at the top right.
yes but i made the critical mistake that i was not logged in, it was commented as a guest, and i can’t edit that
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