How do we actually define a function with cardinals as domain? For example, take domain of the function as $\aleph_1$. Do we define it for all cardinal numbers strictly less than $\aleph_1$?
2026-03-28 20:54:06.1774731246
Functions with cardinals as domain
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There are two fine points here.
Cardinals are ordinals (in the context of $\aleph$ numbers anyway). When we say that we define a function on $\aleph_1$ it might be the case that we define a function on the cardinals below $\aleph_1$, or it might be the case that we are interested in all the ordinals below $\aleph_1$.
Cardinals are sets. Of course, the collection of all cardinals is not a set. But each cardinal is a set. So we can define a function on that set very much like we define a function on any other set.
One can use the inherent structure of cardinals and ordinals, or one can ignore that, and give a different structure if needs be.
And remember that there are many functions on an infinite set, and those need not be definable. They simply exist out there, in the set theoretical universe.