*Throughout, I work in $\mathsf{MK}$ in order to be able to conveniently quantify over logics; if one prefers, we can restrict attention to (say) $\Sigma_{17}$-definable logics and work in $\mathsf{ZFC}$. By "logic" I mean "regular logic containing $\mathsf{FOL}$ and having countably many formulas in a finite language" (for example, $\mathsf{SOL}$).*

Given a logic $\mathcal{L}$, consider the following two $\mathcal{L}$-theories $\mathscr{ZFC}(\mathcal{L})$ and $\mathscr{M}(\mathcal{L})$ defined as follows. We let $\mathscr{ZFC}(\mathcal{L})$ be the $\mathcal{L}$-theory consisting of

the "boring" $\mathsf{ZFC}$-axioms Pairing, Union, Infinity, Choice, Regularity, and Extensionality, and

the Separation and Replacement schemes modified to allow formulas coming from $\mathcal{L}$.

Note that $\mathscr{ZFC}(\mathsf{SOL})$ is not quite the same thing as "second-order $\mathsf{ZFC}$." Meanwhile, $\mathscr{M}(\mathcal{L})$ is $\mathscr{ZFC}(\mathcal{L})$ **plus** for each $\mathcal{L}$-formula $\varphi(x)$ the reflection instance $$\forall x[\varphi(x)\rightarrow\exists \alpha(x\in V_\alpha\wedge \varphi(x)^{V_\alpha})].$$

It's easy to state and prove in (first-order!) $\mathsf{MK}$ that $V_\alpha\models\mathscr{ZFC}(\mathcal{L})$ for every logic $\mathcal{L}$ whenever $\alpha$ is inaccessible; somewhat conversely, assuming $\mathsf{V=L}$ this is optimal already for $\mathcal{L}=\mathsf{SOL}$ since $V_\alpha\models_\mathsf{SOL}\mathscr{ZFC}(\mathsf{SOL})$ only if $\alpha$ is $L$-inaccessible.

The situation for $\mathscr{M}$ is more complicated. For example, if $\alpha$ is the least inaccessible then $V_\alpha\not\models_\mathsf{SOL}\mathscr{M}(\mathsf{SOL})$ since the sentence "$\mathsf{Ord}$ is inaccessible" is second-order expressible and holds in $V_\alpha$ but not in any smaller $V_\beta$. Instead, we need to go a bit higher. Say that a cardinal $\kappa$ is *chromatic* iff $\kappa$ is an inaccessible limit of inaccessibles and the following holds (letting $I$ be the set of inaccessibles $\le\kappa$):

For every family $C=(c_i)_{i\in\omega}$ of $2$-colorings $c_i: [I]^2\rightarrow 2$, there is some $\lambda\le\kappa$ such that for all inaccessible $\alpha<\lambda$ and all $i\in\omega$ with $c_i(\{\alpha,\lambda\})=1$ there is some $\beta\in(\alpha,\lambda)$ with $c_i(\{\alpha,\beta\})=1$.

If $\kappa$ is a chromatic cardinal, then for every countable logic $\mathcal{L}$ there is a $\lambda<\kappa$ such that $V_\lambda\models_\mathcal{L}\mathscr{M}(\mathcal{L})$. Basically, for a given $\mathcal{L}$ with formulas $(\varphi_i)_{i\in\omega}$ and inaccessible cardinals $\alpha<\beta<\kappa$ we let $c_i(\{\alpha,\beta\})=1$ iff $V_\beta\models\varphi(\alpha)$. And essentially trivially, this is optimal.

My first question is about chromaticity itself:

Question 1: What are chromatic cardinals in more familiar language?

I suspect chromaticity is *much* weaker than Mahlo-ness, but I don't immediately see how to prove that.

My second question is about the specific strength of the second-order analogue of $\mathsf{ZFC}$ + reflection as indicated above:

Question 2: What is the consistency strength of "There is an inaccessible $\alpha$ such that $V_\alpha\models_\mathsf{SOL}\mathscr{M}(\mathsf{SOL})$"?

It's easy to see that this is weaker than the existence of a chromatic cardinal; I'm interested in getting a better sense of how much weaker it is.

second-orderformulas. Right? But I still fail to see why they're not equivalent: doesn't second-order comprehension ensure that anything we can write with a second-order formula define a second-order object, and conversely, any second-order object can be considered as a single-variable formula? $\endgroup$5more comments