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Here is a description of cardinal number.

Cardinal numbers are identified with the set of ordinals preceding them.

Is this OK?

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@Martin: Do you think that maybe [definition] or [terminology] fit this question? I was going to add one, and then I hesitated. – Asaf Karagila Apr 19 '13 at 19:36
@Asaf If I had to choose one from the two tags you've suggested, I would go with definition. (Based on the content of the tag-wikis for these two tags.) – Martin Sleziak Apr 19 '13 at 19:40
up vote 4 down vote accepted

It would be better to say:

Ordinals are von Neumann ordinals: each ordinal is the set of its predecessors, and cardinal numbers are identified with initial ordinals.

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Thanks Brain. Could you help me to answer… – Paul Apr 19 '13 at 6:31
@Paul: I’ll take a look, but it may be a while before I can give it serious thought. – Brian M. Scott Apr 19 '13 at 6:33
@Brain: It doesn't matter. I'll wait for you. – Paul Apr 19 '13 at 6:37

Your statement is unclear, or even wrong, because every ordinal is identified as the set of preceding ordinals. It is also unclear whether or not you mean that every cardinal number is identified with an ordinal, or that every ordinal which can be identified with the set of its predecessors is a cardinal.

Cardinal numbers are identified with initial ordinals, which are ordinals that there is no injection from them into a smaller ordinal.

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