# How many steps does it take to "Tarski-Vaughtify" second-order logic?

Given a regular logic $$\mathcal{L}$$, let $$\preccurlyeq_\mathcal{L}$$ be the usual elementary submodelhood relation for $$\mathcal{L}$$. There is also a separate submodelhood relation coming from the Tarski-Vaught test: say that $$\mathfrak{A}\trianglelefteq_\mathcal{L}\mathfrak{B}$$ if $$\mathfrak{A}$$ is a substructure of $$\mathfrak{B}$$ and for every $$\mathcal{L}$$-formula $$\varphi$$ with parameters from $$\mathfrak{A}$$ we have $$\varphi^\mathfrak{B}\not=\emptyset\implies \varphi^\mathfrak{B}\cap\mathfrak{A}^{arity(\varphi)}\not=\emptyset.$$

Usually $$\trianglelefteq_\mathcal{L}$$ is strictly between $$\subseteq$$ and $$\preccurlyeq_\mathcal{L}$$. Now every logic $$\mathcal{L}$$ has a distinguished fragment which "plays nicely" with this substructurehood relation, namely $$\mathcal{L}_{TV}:=\{\varphi\in\mathcal{L}: \mathfrak{A}\trianglelefteq_\mathcal{L}\mathfrak{B}\implies\varphi^\mathfrak{A}=\varphi^\mathfrak{B}\cap\mathfrak{A}^{arity(\varphi)}\}.$$ In general we need not have $$\mathcal{L}_{TV}=(\mathcal{L}_{TV})_{TV}$$, and so we can non-boringly iterate this process through the ordinals:

• $$\mathcal{L}_{TV(0)}=\mathcal{L}$$,

• $$\mathcal{L}_{TV(\alpha+1)}=(\mathcal{L}_{TV(\alpha)})_{TV}$$, and

• $$\mathcal{L}_{TV(\lambda)}=\bigcap_{\alpha<\lambda}\mathcal{L}_{TV(\alpha)}$$ for $$\lambda$$ limit.

I've asked at MSE about one aspect of this process applied to second-order logic for a single step; here I'd like to ask an orthogonal question. Let the TV-number of a logic $$\mathcal{L}$$ be the least $$\gamma$$ such that $$\mathcal{L}_{TV(\gamma)}=\mathcal{L}_{TV(\gamma+1)}$$.

What is the TV-number of second-order logic?

Embarrassingly, as far as I know we could have $$\gamma=1$$ (we can't have $$\gamma=0$$ since $$\mathsf{SOL}_{TV}$$ has the full downward Lowenheim-Skolem property). I'd separately be interested in any references about this construction; the abstract model theory literature is messy, and I'd like to avoid reinventing as many wheels as possible.

Motivation: The $$TV$$-process is basically a quick way of whipping up logics with the downward Lowenheim-Skolem property or analogues thereof (count the formulas and think about Skolem functions), but given that it's not idempotent the "$$TV^\infty$$-process" $$\mathcal{L}\leadsto \bigcap_{\alpha\in \mathit{Ord}}\mathcal{L}_{TV(\alpha)}$$ may actually be a more natural way of producing such logics. I'm especially interested in second-order logic and its fragments, and so this question is basically a step towards figuring out which of $$TV$$ or $$TV^\infty$$ I should be thinking about at the moment.

• I was going to ask about iterating the $TV$ operator on your other question, but I never got around to it. Mar 15, 2021 at 0:01
• I hereby propose that the process be called "televise" as in "Noah televised second-order logic". Mar 17, 2021 at 8:05