Talk:Class of All Ordinals is Ordinal

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

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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)