Is there any literature, especially when doing mathematics without the axiom of choice, that discusses using collections of cardinal numbers in place of individual cardinal numbers, when discussing cardinal numbers with certain properties?

That's a little vague, and I will presently give a few motivational examples that should help to clarify what I'm thinking of. But first, a quick definition of the translation between cardinal numbers and collections of cardinal numbers assuming the axiom of choice: If *K* is a cardinal number, then {*L* | *L* < *K*} is a collection of cardinals which is small and downward-closed. Conversely, if *C* is a collection of cardinals which is small and downward-closed, then there exists a unique cardinal *K* (to wit, the smallest cardinal that does not belong to *C*) such that *C* = {*L* | *L* < *K*}. So the correspondence between these two perspectives is straightforward —*if* we assume choice.

Now I'll consider some types of cardinal numbers. A *weak limit cardinal* is an infinite cardinal *K* such that *L*^{+} < *K* whenever *L* < *K*. A *strong limit cardinal* is an infinite cardinal *K* such that 2^{L} < *K* whenever *L* < *K*. A *regular cardinal* is an infinite cardinal *K* such that Σ_{i ∈ I} *L*_{i} < *K* whenever |*I*| < *K* and each *L*_{i} < *K*. An *inaccessible cardinal* is an uncountable regular limit cardinal. A *weakly compact cardinal* is an inaccessible cardinal *K* such that the height of a tree is less than *K* whenever every level has width less than *K* and every branch has length less than *K* (the tree property). Etc.

Although these are all properties of an individual cardinal number *K*, they all refer to that cardinal only through the collection of smaller cardinal numbers. (The exceptions are the adjectives ‘infinite’ and ‘uncountable’, which are really only there to rule out trivial cases.)

If you don't assume the axiom of choice, then it's easy to still consider these conditions on a small, downward-closed collection of cardinals, but now this is more general than a collection of the form {*L* | *L* < *K*}. If you go beyond doubting the axiom of choice and drop the law of excluded middle (so doing constructive mathematics, in a moderate sense), you can get more flexibility by messing with the interpretation of ‘downward-closed’; for example, the collection {0, 1, 2, …} of all finite (in the strictest sense) cardinal numbers is closed under taking decidable sub-cardinals but not arbitrary sub-cardinals and so gives a ‘regular’ collection of cardinals which is different from {*L* | *L* < ℵ_{0}}.

I'm interested in understanding what regular cardinals and inaccessible cardinals should be in constructive mathematics. (Bigger than that, I don't really even understand them in classical mathematics.) It seems to me that they have to be collections of cardinals rather than individual cardinal numbers. But this perspective already makes sense in classical mathematics, especially without the axiom of choice. Has anybody studied this?

allinfinite cardinals satisfy your definition of "weak limit cardinal". – Ricky Demer Dec 22 '10 at 9:47ordinals, notcardinalsas previously in the paragraph. – Mike Shulman Dec 22 '10 at 20:07