I was wondering about a way to make really large countable ordinals. It turns out that in certain models of ZFC (for example pointwise definable ones) every ordinal is definable with *no* parameters, which means you can't expect your ordinal to exist in every model of ZFC without having some FOST (language of first-order set theory) equivalent characterization.

Of course, one can simply make an axiom (a scheme with constant a symbol) to create such an ordinal which is not definable within ZFC (unless something with $\omega$-consistency, I don’t know what really). One can then make a scheme with more constant symbols that are all ordinals which cannot be defined from previous ordinals.

A sort of "limit" of this is making an axiom stating the existence of an elementary embedding from $\omega_1$ to itself with a critical point. I believe this could quite easily be proven inconsistent, but I'm not sure how to start.

I call this "The Axiom of Countable Ineffability" or $\text{CI}$ for short. Here's the formal definition of $\text{CI}(Q)$ for a set of formulae $Q$:

- There is some ordinal $\alpha\in\omega_1$ such that $j(\alpha)\neq\alpha$
- For every $\alpha\in\omega_1$, $j(\alpha)\in\omega_1$.
- For every $\varphi\in Q$, an axiom stating that for any $\alpha_0,\alpha_1...\in\omega_1$ with $\varphi^{\omega_1}(\alpha_0,\alpha_1...)$, $\varphi^{\omega_1}(j(\alpha_0),j(\alpha_1)...)$

Of course, because it only quantifies over $\varphi^{\omega_1}$ for FOST $\varphi$, for which there exists a set-sized truth predicate in $V$ (defined on the set of all finite sequences of countable ordinals), **this can be stated as a single FOST axiom (as long as the class $\{\varphi^{\omega_1}:\varphi\in Q\}$).** In fact, this is true for any uncountable ordinal, not just $\omega_1$.

**The Questions**

- What is the consistency of $\text{ZFC}+\text{CI}(\Sigma_n)$?
- What is the consistency of $\text{ZFC}+\text{CI}$?