# Is there an abstract logic that defines the mantle?

It is a known result by Scott and Myhill that the second-order version of $$L$$ yields $$\mathrm{HOD}$$. Recently, Kennedy, Magidor, and Väänänen (Inner models from extended logics: Part I and II) investigated inner models given by logics with generalized quantifiers, which yields a logic intermediate between first-order and second-order logic. It motivates the following question:

Is there a logic that produces the mantle?

(Here the choice of the mantle is somewhat arbitrary; we may replace it by 'generic mantle', 'symmetric mantle' or whatever. I will focus on the mantle in this question, but I welcome discussing other cases.)

Of course, the answer is trivial if we assume, like $$V=L$$ or $$V=L[G]$$ for some $$L$$-generic $$G$$. I want to ask the existence of logic which defines the mantle uniform to models of ZFC.

Is there a (ZFC-definable) abstract logic $$\mathcal{L}$$ such that the inner model given by $$\mathcal{L}$$ is (ZFC-provably) the mantle?

(Under model-theoretic terms, is there $$\mathcal{L}$$ such that for any model $$M$$ of $$\mathsf{ZFC}$$, the inner model given by $$\mathcal{L}$$ is the mantle of $$M$$?)

Here are some of my rough thoughts:

• Sublogics of higher-order logics are not the candidate for $$\mathcal{L}$$: the corresponding inner models of higher-order logics are $$\mathrm{HOD}$$ (if my reasoning is correct), so the sublogics yield a submodel of $$\mathrm{HOD}$$. However, $$\mathrm{HOD}$$ need not be the mantle. (Theorem 70 of Fuchs, Hamkins, and Reitz (Set-theoretic geology).)

• We can rule out $$\mathcal{L}_{\kappa\kappa}$$, which yields Chang model. The inner model given by $$\mathcal{L}_{\kappa\kappa}$$ is the least transitive model of ZF that contains all ordinals and is closed under $$<\kappa$$-sequences (Theorem II of Chang's Sets constructible using $$L_{\kappa\kappa}$$.) However, the mantle need not be closed under $$<\kappa$$-sequences. (A generic extension of $$L$$ would be an example.)

• An observation: if the mantle has the form $L[A]$, where $A$ is a class of ordinals, then we may define an artificial $A$-recovering quantifier $Q^A$ as follows: $N\vDash (Q^A xy)\varphi(x,y,\vec{a})$ iff $\{(x,y)\in N^2\mid N\vDash \varphi(x,y,\vec{a})\}$ is a linear order of ordertype in $A$. Then the resulting model is just $L[A]$. – Jason Zesheng Chen Jul 25 at 0:04
• Can you be more specific about what you mean by an abstract logic? It is hard to find the definition online, or more accurately, it is easy to find a number of inequivalent definitions. It seems to me it might be possible to realize any inner model $M$ as $C(\mathcal L)$ for some proper class abstract logic $\mathcal L$ which is ad hoc yet definable from the predicate $M$. – Gabe Goldberg Jul 26 at 22:09
• @GabeGolsberg Thank you for your comment. I did not know that it has an ambiguous definition (I initially assumed the definition in Model Theory by Chang and Keisler.) – Hanul Jeon Jul 27 at 2:21
• What if $\mathcal L$ has formulas $\varphi_x$ for every $x\in M$ such that $\varphi_x$ is false unless $\mathfrak{A}$ is a structure with one relation symbol and for some transitive $y\in M$, there is an isomorphism $f : \mathfrak{A}\to (y,\in)$ such that $f(x)\in y$. In other words, $\mathcal L$ has formulas coding all elements of $M$. It seems like by induction, the $\alpha$-th level of the $C(\mathcal L)$-hierarchy is $(V_\alpha)^M$. I worry that I am misunderstanding the definition of an abstract logic. – Gabe Goldberg Jul 27 at 4:28
• @GabeGoldberg Your argument seems relevant to Hamkins' previous answer. – Hanul Jeon Jul 27 at 9:12

Combining Goldberg's comment and Hamkins' answer seems to work. Especially, for any inner model $$M$$ of ZF, we have an abstract logic $$\mathcal{L}$$ whose corresponding inner model $$L^\mathcal{L}$$ is $$M$$.
Consider the sublogic of $$\mathcal{L}_{\infty,\omega}$$ such that infinite conjunction and disjunctions are only allowed to set of formulas in $$M$$. In fact, $$\mathcal{L}=\mathcal{L}_{\infty,\omega}^M$$.
Define $$\psi_A$$ for $$A\in M$$ as Hamkins defined: to repeat the definition, $$\psi_A(x):= \bigvee_{a\in A} (\forall v : v\in u\leftrightarrow \psi_a(u)).$$ Then $$\psi_A(x)$$ is a member of $$M$$ by induction on $$A\in M$$.
We can see that if $$A\in M$$, $$A\subseteq V_\alpha^M$$ then $$A=\{u\in V^M_\alpha \mid V^M_\alpha\models \psi_A(u)\}.$$
Hence the $$\alpha$$th hierarchy $$L_\alpha^\mathcal{L}$$ contains $$V^M_\alpha$$ (It can be shown by induction on $$\alpha$$.) Therefore $$M\subseteq L^\mathcal{L}$$. On the other hand, an inductive argument shows that the $$\alpha$$th hierarchy $$L^\mathcal{L}_\alpha$$ is a member of $$M$$ (we need the absoluteness of the satisfaction relation for $$\mathcal{L}$$ between $$M$$ and $$V$$), so $$L^\mathcal{L}\subseteq M$$.
• Of course, this is a proper-class-sized logic - it would be quite reasonable to ask for a set-sized logic, that is, one whose class of $\Sigma$-formulas is a set for each language $\Sigma$. And then I see no similar trick. – Noah Schweber Jul 27 at 23:46
• In that case, it seems like the $C(\mathcal L)$ must be contained in $\text{HOD}_{\text{Ord}^\kappa,S}$ for some set $S$ (corresponding to the set of sentences) and some cardinal $\kappa$ (corresponding to the number of free variables allowed in a single sentence sentence). I bet you can get a model where the mantle is not contained in $\text{HOD}_{\text{Ord}^\kappa,S}$ for any $\kappa$ and $S$. It may be interesting to look instead at logics that have a Lowenheim number. – Gabe Goldberg Jul 28 at 0:06