Definability and Models of a first-order theory of cardinals

Question

Let $\mathcal{M}=(\mathrm{Card},<,x \mapsto \aleph_x)$. Also, have $C_0 = \{\kappa \ |\ \aleph_\kappa = \kappa\}$, and $C_{n+1} = \{\kappa \ |\ \kappa = \sup (C_n \cap \kappa)\}$. One can see quite easily that arbitrarily large elements of any $C_n$ are definable in $\mathcal{M}$. Thus, a question arises: Is the least cardinal $\kappa$ that is closed under the operations imposed by each $C_n$ elementarily equivalent to $\mathcal{M}$, and is the existence of such a cardinal provable in ZFC?

Answer 1

Here is a partial answer, this cardinal exists. It is the $\omega^\omega$th fixed point. To see why, note that $C_0$ is just the class of fixed points, and $C_1$ is the limits of fixed points (which are themselves fixed points, easily). So the least member of $C_1$ is the $\omega$th fixed point. Similarly, $C_2$ is the limit of limits of fixed points, so the least such cardinal is the $\omega^2$ fixed point, etc.

So the cardinal you're looking for is indeed the $\omega^\omega$th fixed point.