Talk:Class of All Ordinals is Ordinal

Copied from Burali-Forti Paradox, but it's an important result, and will be used later. --asalmon

Blah
Yes, this can be so. Or not. Kelley says yes, Smullyan &amp; Fitting say no. Kelley defines "ordinal" so that $\operatorname{On}$ is one, but then defines "ordinal number" as an element of $\operatorname{On}$ (and proves that $\operatorname{On}$ is the only ordinal which is not a set). This approach certainly allows for certain things to be simpler than they might otherwise be, but we have to decide one way or another. --Dfeuer (talk) 23:40, 5 April 2013 (UTC)


 * Tricky.


 * Without having studied this area of maths, I will make a guess: it appears that this may be a Definition:Philosophical Position like Law of Excluded Middle or Axiom:Axiom of Choice, or some such: the question is voiced: Is the ordinal class an ordinal?


 * OTOH if this is "proved" from the definitions given, then those need to be studied, and the precise chain of thought leading to this needs to be examined to see which one is breakable so as to allow either Kelley's or S&F's approach. The philosophical position may need to be established "further back" from this. --prime mover (talk) 07:39, 6 April 2013 (UTC)


 * It probably is a philosophical thing. What I like about Kelley's approach is that it cleanly supports both concepts. Either way of doing it is fine, but we currently have some pages relying on one definition and other pages relying on the other, which is much less fine. Unless someone can determine that a particular approach is now universal among modern texts, I would propose implementing Kelley's approach (ordinal/ordinal number) and letting me slog through the couple hundred pages this affects to figure out which needs which. --Dfeuer (talk) 13:12, 6 April 2013 (UTC)

Invalid proof and statement issues
Invalid: Even if it really made sense to talk about the initial segment determined by $\operatorname{On}$, it's not actually even relevant. The final appeal to the definition of an ordinal doesn't work—Definition 3 requires that each element of an ordinal equals its initial segment, not that an upper bound does so.

Statement issues: Since L_F has determined that it is best NOT to consider $\operatorname{On}$ an ordinal, and he is more qualified than I to make that determination, the theorem statement and title obviously needs to be modified. My preference would probably be to make up a name like "Ordinal-Like Class", essentially following Kelley who used "ordinal" for that concept and "ordinal number" for ordinals that are sets. The other option (messier) would be to have multiple theorem statements here, one for each definition of ordinal, to show that the ordinal class satisfies all but the set condition of each. --Dfeuer (talk) 00:02, 25 April 2013 (UTC)


 * This is completely false. The problem starts with the very definition of ordinal number. An ordinal number must be a set. Despite the fact that $\operatorname{On}$ behaves as a transitive and well-ordered set, it cannot be an ordinal, since The Burali-Forti paradox states that $\operatorname{On}$ is not a set. --Erickgrm (talk) 12:27, 21 May 2013 (UTC)


 * The difficulty here, Erickgrm, is that different texts define an ordinal differently. For example, both Gödel and Kelley define an ordinal as a transitive class strictly well-ordered by epsilon, and an ordinal number as an ordinal which is a set (or, in Kelley's case, an element of $\operatorname{On}$, which amounts to the same thing). I've been asking the powers that be here for a resolution to this issue (either adopting the Gödel/Kelley approach or coming up with a name for an ordinal-like class), but I have not managed to get that done. --Dfeuer (talk) 17:38, 21 May 2013 (UTC)

Restructuring
As for my 2 unsolicited cents, I think this can be most elegantly solved by an "also defined as" on Definition:Ordinal and potentially also on Definition:Class of All Ordinals but the latter is a bonus.

All the proofs can also be phrased in the existing language without much extra wordage, but then also without the added confusion of "ordinal" vs "ordinal number". I don't consider this of the order of magnitude of AoC or LEM at all. &mdash; Lord_Farin (talk) 20:59, 27 June 2022 (UTC)


 * That's sort of my proposed approach. I'm building up to it. --prime mover (talk) 22:07, 27 June 2022 (UTC)