Talk:Cardinal Number Less than Ordinal

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Can this be revisited? The result being proved seems to be a bit different from the statement. --prime mover (talk) 18:17, 31 August 2012 (UTC)

Good catch. --Andrew Salmon (talk) 18:51, 31 August 2012 (UTC)

$x \in T \implies \bigcap T \subseteq x$ is precisely what Intersection is Subset (the theorem referenced on that line) says. --Andrew Salmon (talk) 21:09, 31 August 2012 (UTC)

Yes indeed, but the issue here is that $\le$ is being used throughout this entire field of mathematics as a synomym for $\subseteq$. While that is indeed true for ordinals, as this is one of the important defining features of an ordinal, it is IMO a mistake to take it for granted in a page without linking to that very fact. Otherwise the most important step of the proof has been omitted.
And while I'm about it, "less than" in the title does not match the "\le" in the proof. Perhaps "not greater than" would be better, unless there's a definitional variant as yet unreferenced such that "less than" by default includes equality. --prime mover (talk) 06:14, 1 September 2012 (UTC)
IMHO this slight abuse of language is justified by the resulting simple title. --Lord_Farin (talk) 08:13, 1 September 2012 (UTC)
I'm not convinced on that point, but I will concede if I am outnumbered. :-) --prime mover (talk) 08:24, 1 September 2012 (UTC)

On a side note, for consistency I think it's best to refer to Ordinal rather than Ordinal Number. --Lord_Farin (talk) 21:26, 31 August 2012 (UTC)