Definition talk:Ordinal

"We will introduce the notation $ Ord(S) $ to denote that $S$ is an ordinal."

I'm dead against this style of notation because it makes mathematics much more difficult to read.

If a particular font is used (even it it's standard math roman) then it's better, but as it is, math italic is used for variables. So until the style of this system of abbreviation is completely rationalised and made readable (and perhaps cited by an authority), it's probably best if it were not used. --prime mover 07:23, 26 November 2011 (CST)

Class or set?
There seems to be a mismatch here... This page defines an ordinal as a set, but Ordinal Class is Ordinal is up here. Which definition is preferred? --abcxyz 23:45, 26 June 2012 (UTC)


 * Ordinal Class is Ordinal refers to the class of all ordinal numbers. That is a proper class (not a set). Andrew Salmon 00:21, 27 June 2012 (UTC)


 * Yes, but then, according to the definition on this page (which requires an ordinal to be a set), you can't say that the ordinal class (which is a proper class) is an ordinal. So my question is which one is preferred: defining an ordinal as a class, or as a set? --abcxyz 00:48, 27 June 2012 (UTC)


 * We shall call an ordinal a class, and a member of the class of all ordinals a set. Andrew Salmon 00:08, 10 July 2012 (UTC)

When explicating such is desired, why not simply say 'ordinal set' (or 'set ordinal', for that matter)? Of course, the 'class of all ordinals' cannot contain itself, and is in fact the 'class of all ordinal sets'. --Lord_Farin 08:52, 10 July 2012 (UTC)

Refactor
I took a stab at separating the melded definitions that were here before and added another one. Obviously we need to figure out which source uses what. --Dfeuer (talk) 17:36, 21 April 2013 (UTC)


 * Substandard. --prime mover (talk) 18:23, 21 April 2013 (UTC)


 * Perhaps, but it was a worse mess before. I'm working on cleaning up some of the very shaky proofs that underly basic properties of ordinals. --Dfeuer (talk) 18:40, 21 April 2013 (UTC)


 * What was there before has been relegated to "Alternative Definition". WTF??? What's wrong with "Definition 3"? Is it genuinely not an equivalent definition?


 * Please remember to add the SourceReview template, and learn how it is used. --prime mover (talk) 19:08, 21 April 2013 (UTC)


 * Please review what was there before. The reason I called one of the options alternative is that it defines an ordinal as an ordered set with a cetain propery, rather than the underlying set thereof. That is very similar, but not actually equivalent, per se. --Dfeuer (talk) 19:31, 21 April 2013 (UTC)


 * The definitions are entirely equivalent. Your distinction is spurious. --prime mover (talk) 20:32, 21 April 2013 (UTC)

Equivalence of Definitions
I propose the following changes:


 * The first is to change $\Epsilon$ to $\Epsilon \! \restriction_S$ in Definition 1, because not only is the epsilon relation a proper class, but this new definition makes more sense because $\Epsilon \! \restriction_S$ is an actual relation.

This change will not alter the basic concept of an ordinal. When one uses the fact that an ordinal is well-ordered, or when someone is attempting to prove that a certain set is well-ordered, they are only concerned with the elements of that particular set. So the current theorems regarding ordinals will not have to be extensively changed to accommodate this.


 * Also, shouldn't the ordering in Definition 3 be a strict well-ordering? It seems inconsistent to me that the first definition defines it as a type of set that is strictly well-ordered by $\in$, but the third definition defines it has a type of well-ordered set that is not-strict. I understand that there is a corresponding strict ordering, but still.

I was able to construct an equivalence of definitions proof of this, but it's only compatible with the definitions if these changes are made. Again, the existing theorems regarding the ordinals won't have to be reworked to accommodate for this, just replace $\Epsilon$ with $\Epsilon \! \restriction_S$.

Since the ordinals are a fundamental concept, I want to see if anyone else agrees with me on this. Thanks for reading. --HumblePi (talk) 13:42, 6 May 2017 (EDT)


 * Unless you can find agreement in the literature of class theory, I would advise against it. But I see your point concerning Definition 3 and the well-ordering or strict well-ordering. --prime mover (talk) 17:47, 6 May 2017 (EDT)


 * Okay, I'll change Definition 3 and publish the proof. As for Definition 1, I'll see if I can find a reputable text that defines an ordinal as such. --HumblePi (talk) 19:00, 6 May 2017 (EDT)


 * I found one! In Herbert B. Enderton's book Elements of Set Theory, he states:

A set $A$ is well ordered by epsilon iff the relation:
 * $\in_A \; = \left\{{\left\langle{x, y}\right\rangle \in A \times A \; | \; x \in y}\right\}$

Is a well-ordering on $A$


 * And then he goes on to say that if $\alpha$ is transitive and well-ordered by epsilon, then it's an ordinal.


 * This is the exact same definition that I proposed, and it was the first book I checked too. I've never owned this book because I couldn't afford it, but I heard about it through How to learn Math and Physics by John Baez. What an amazing coincidence! --HumblePi (talk) 20:36, 6 May 2017 (EDT)


 * Good job. Feel free to add that citation in the "Sources" section, using the standard form (if you don't know what that form is, please feel free to copy that which you can find on many of the pages in this website). --prime mover (talk) 05:37, 7 May 2017 (EDT)