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.
Best Answer
The first chromatic cardinal is the first Mahlo cardinal. (Per the connection with the reflection in the question, I assume that in the definition, $α$ and the first argument of $c_i$ need not be inaccessible.) If we allowed $c_i$ for $i≤δ$, then the first chromatic cardinal would be the first Mahlo above $δ$.
If $κ$ is Mahlo, then it is chromatic. Define club $C⊆κ$ with $λ∈C ⇔ ∀α<λ \, ∀i \, \min(\{β>α:c_i(α,β)\})<λ$ where $\min(S)$ is modified to return 0 if $S$ is empty. Then any inaccessible $λ∈C$ satisfies the desired property.
In the other direction, if $κ$ is below the first Mahlo, then there is a one-to-one function that for every inaccessible below $κ$ returns a smaller ordinal. For example, pick a club $D$ that excludes inaccessibles. If $β$ is the least inaccessible above some element of $D$, then set $f(β)=\max(β∩D)$. Let $λ=\min(D\setminus β)$, and if there are inaccessibles in $(β,λ)$, let $λ'$ be largest inaccessible or limit of inaccessibles $<λ$. Set (for example) $f(λ')=β$, and analogously to $D$ pick a club $E⊂λ'$ above $β$ that excludes inaccessibles. Analogously proceed by recursion until all inaccessibles are taken care of.
For your second question, despite the power of second order logic, the cardinals in question are precisely inaccessible $α$ with $V_α ≺_{Σ_{1,V}} L_1(V_{α+1})$, where $Δ_{0,V}$ formulas allow querying about $x$ whether $∃α \, x=V_α$ (for the true $V$), and $L$ is the constructible hierarchy. $Σ_{1,V}$ formulas are upwards absolute, and $L_1$ corresponds to definability, so the above reduces to the desired reflection relation. If there is an inaccessible $λ$ with $V_λ ≺_{Σ_2} V$, then $α$ for the question exists for every parameter-free definable $Δ^V_2$ logic (even if there are no Mahlos in $V$; also, we can allow parameters in $V_λ$).