The statement "$\alpha$ is a cardinal" is not absolute

Question

Jech in his "Set Theory" states without elaboration that cardinal concepts are not absolute generally. I think this is problematic. Any thought?

Answer 1

Consider a countable elementary submodel of $V_\kappa$ for some cardinal $\kappa>\omega_1$.

Since $\omega_1$ is definable there, it is in the submodel. Great, now take a transitive collapse. That is a countable transitive model of some amount of set theory. But now $\omega_1$ was collapsed to a countable ordinal.

So we have a countable ordinal and a transitive set which thinks this ordinal is an uncountable cardinal. So being a cardinal is not absolute.

Answer 2

In fact, I'd go a little further and say that cardinal properties are seldom absolute. Remember, "absolute" means that the property's truth value doesn't change when you move between larger and smaller models.

Even being a cardinal is not absolute. For example, it's fairly straightforward to construct a pair of models $M_0 \subseteq M_1$ so that the ordinal $M_0$ thinks is the first uncountable cardinal is actually countable in $M_1$ (so isn't a cardinal at all).

The issue is that most cardinal properties involve either the phrase "there exists..." or the phrase "for every...", once you navigate through all of the definitions. What that means is that, usually, you can either move down to a smaller model in which the thing that used to exist doesn't, or up to a larger model in which the thing that used to always be true has a counterexample.