# 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)