# Is there an abstract logic that defines the mantle?

MathOverflow Asked on January 3, 2022

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

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 $$Ain M$$ as Hamkins defined: to repeat the definition, $$psi_A(x):= bigvee_{ain A} (forall v : vin uleftrightarrow psi_a(u)).$$ Then $$psi_A(x)$$ is a member of $$M$$ by induction on $$Ain M$$.

We can see that if $$Ain M$$, $$Asubseteq V_alpha^M$$ then $$A={uin V^M_alpha mid V^M_alphamodels psi_A(u)}.$$

Hence the $$alpha$$th hierarchy $$L_alpha^mathcal{L}$$ contains $$V^M_alpha$$ (It can be shown by induction on $$alpha$$.) Therefore $$Msubseteq L^mathcal{L}$$. 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 $$L^mathcal{L}subseteq M$$.

Answered by Hanul Jeon on January 3, 2022