Re: Cardinality

In the first chapter of his graduate text on Set Theory, Kunen defines a cardinal as an ordinal which is not equipollent to a strictly smaller ordinal He also defines aleph_a and omega_a (where a is an ordinal) to be alternative notations for the same thing; these represent specific cardinals, hence ordinals. Hence, with Kunen's notation, it does indeed make sense to speak of aleph_(aleph_a), which one could also represent by omega_(omega_a) or aleph_(omega_a).

It is a theorem of ZF that aleph_a exists for every ordinal a. That is, one does not need the Axiom of Choice. See Kunen for a proof. (Basically, given an ordinal a, one well-orders the well-orderings of a to get a cardinal strictly greater than a.)

If one is working in ZF, one might not want to use Kunen's definition of a cardinal Even so, each of Kunen's cardinals is representative of a distinct cardinal in the definition one would likely use.


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Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan

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