(The original version of this question was much narrower and less natural; but see the edit history if interested.)

Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ and having the finite use property and the strong downward Lowenheim-Skolem property together with, for each finite language $\Sigma$, an injection $i_\Sigma$ from the set of sentences $\mathcal{L}[\Sigma]$ to $\omega$ (I'll abbreviate $i=\{i_\Sigma:\Sigma$ a finite language$\}$). Using $i$ to conflate sentences with naturals appropriately, we can define an inner model $$M_\mathcal{L}^r:=L[r,\models_\mathcal{L}]$$ for each real $r$. Strong dLS (= "For every $X\subseteq\mathfrak{A}$ in a finite language there is a $\mathfrak{B}$ with $\vert\mathfrak{B}\vert\le X\cdot\aleph_0$ and $X\subseteq\mathfrak{B}\preccurlyeq_\mathcal{L}\mathfrak{A}$") ensures that we have a nice condensation phenomenon in the $M_\mathcal{L}^r$s, at least once $r$ is "strong enough" to code the relevant basic information, so these are in my opinion reasonably natural modifications of $L$ to consider.

Intuitively, due to the complexity of $i$ we might need the real $r$ to "unpack" things appropriately; for example, if we take $\mathcal{L}$ to be the "Tarski-Vaughtification" of second-order logic (see e.g. here) and use the obvious $i$ then we can really only hope for good behavior if we're given a real coding which second-order sentences are actually in $\mathcal{L}$. For this reason, I'm going to focus on the behavior of these models on a cone:

Definition: The ideal inner model theory of a good logic $\mathcal{L}$ is the first-order theory $$\mathsf{IIMT}_\mathcal{L}:=\{\varphi\in\mathsf{FOL}[\{\in\}]: \exists r\forall s\ge_Tr M_\mathcal{L}^s\models\varphi\}.$$

Due to the on-a-cone focus, the specific "coding map" $i$ used will play no role.

A secondary question focuses on combinatorics; here, I'm going to ask more specifically about large cardinals.

Question: Is it consistent (relative to large cardinals) that there is a good logic $\mathcal{L}$ such that $\mathsf{IIMT}_\mathcal{L}$ contains "There is a measurable cardinal?"

(The original version of this question was much narrower and less natural; but see the edit history if interested.)

Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ and having the finite use property and the strong downward Lowenheim-Skolem property together with, for each finite language $\Sigma$, an injection $i_\Sigma$ from the set of sentences $\mathcal{L}[\Sigma]$ to $\omega$ (I'll abbreviate $i=\{i_\Sigma:\Sigma$ a finite language$\}$). Using $i$ to conflate sentences with naturals appropriately, we can define an inner model $$M_\mathcal{L}^r:=L[r,\models_\mathcal{L}]$$ for each real $r$. Strong dLS (= "For every $X\subseteq\mathfrak{A}$ in a finite language there is a $\mathfrak{B}$ with $\vert\mathfrak{B}\vert\le X\cdot\aleph_0$ and $X\subseteq\mathfrak{B}\preccurlyeq_\mathcal{L}\mathfrak{A}$") ensures that we have a nice condensation phenomenon in the $M_\mathcal{L}^r$s, at least once $r$ is "strong enough" to code the relevant basic information, so these are in my opinion reasonably natural modifications of $L$ to consider.

Intuitively, due to the complexity of $i$ we might need the real $r$ to "unpack" things appropriately; for example, if we take $\mathcal{L}$ to be the "Tarski-Vaughtification" of second-order logic (see e.g. here) and use the obvious $i$ then we can really only hope for good behavior if we're given a real coding which second-order sentences are actually in $\mathcal{L}$. For this reason, I'm going to focus on the behavior of these models on a cone:

Definition: The ideal inner model theory of a good logic $\mathcal{L}$ is the first-order theory $$\mathsf{IIMT}_\mathcal{L}:=\{\varphi\in\mathsf{FOL}[\{\in\}]: \exists r\forall s\ge_Tr M_\mathcal{L}^s\models\varphi\}.$$

Due to the on-a-cone focus, the specific "coding map" $i$ used will play no role.

A secondary question focuses on combinatorics; here, I'm going to ask more specifically about large cardinals.

Question: Is it consistent (relative to large cardinals) that there is a good logic $\mathcal{L}$ such that $\mathsf{IIMT}_\mathcal{L}$ contains "There is a measurable cardinal?"

Note that it is crucial that we ask about an actual measurable cardinal, rather than merely an inner model with a measurable: already the logic $\mathsf{SOL}^{TV}$ mentioned above gives us every projective real, which under large cardinal assumptions (if memory serves) gives rise to very strong inner models.

(The original version of this question was much narrower and less natural; but see the edit history if interested.)

Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ and having the finite use property and the strong downward Lowenheim-Skolem property together with, for each finite language $\Sigma$, an injection $i_\Sigma$ from the set of sentences $\mathcal{L}[\Sigma]$ to $\omega$. Letting $i=\{i_\Sigma:\Sigma$ a finite language$\}$, we can define an inner model $$M_\mathcal{L}^r:=L[\{\langle\mathfrak{A},n\rangle: \mathfrak{A}\models_\mathcal{L}i_{\mathit{Lang}(\mathfrak{A})}^{-1}(n)\},r]$$ for each real $r$. Strong dLS (= "For every $X\subseteq\mathfrak{A}$ in a finite language there is a $\mathfrak{B}$ with $\vert\mathfrak{B}\vert\le X\cdot\aleph_0$ and $X\subseteq\mathfrak{B}\preccurlyeq_\mathcal{L}\mathfrak{A}$") ensures that we have a nice condensation phenomenon in the $M_\mathcal{L}^r$s, at least once $r$ is "strong enough" to code the relevant basic information, so these are in my opinion reasonably natural modifications of $L$ to consider.

Intuitively, due to the complexity of $i$ we might need the real $r$ to "unpack" things appropriately; for example, if we take $\mathcal{L}$ to be the "Tarski-Vaughtification" of second-order logic (see e.g. here) and use the obvious $i$ then we can really only hope for good behavior if we're given a real coding which second-order sentences are actually in $\mathcal{L}$. For this reason, I'm going to focus on the behavior of these models on a cone:

Definition: The ideal inner model theory of a good logic $\mathcal{L}$ is the first-order theory $$\mathsf{IIMT}_\mathcal{L}:=\{\varphi\in\mathsf{FOL}[\{\in\}]: \exists r\forall s\ge_Tr M_\mathcal{L}^s\models\varphi\}.$$

Due to the on-a-cone focus, the specific "coding map" $i$ used will play no role.

A secondary question focuses on combinatorics; here, I'm going to ask more specifically about large cardinals.

Question: Is it consistent (relative to large cardinals) that there is a good logic $\mathcal{L}$ such that $\mathsf{IIMT}_\mathcal{L}$ contains "There is a measurable cardinal?"

(The original version of this question was much narrower and less natural; but see the edit history if interested.)

Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ and having the finite use property and the strong downward Lowenheim-Skolem property together with, for each finite language $\Sigma$, an injection $i_\Sigma$ from the set of sentences $\mathcal{L}[\Sigma]$ to $\omega$ (I'll abbreviate $i=\{i_\Sigma:\Sigma$ a finite language$\}$). Using $i$ to conflate sentences with naturals appropriately, we can define an inner model $$M_\mathcal{L}^r:=L[r,\models_\mathcal{L}]$$ for each real $r$. Strong dLS (= "For every $X\subseteq\mathfrak{A}$ in a finite language there is a $\mathfrak{B}$ with $\vert\mathfrak{B}\vert\le X\cdot\aleph_0$ and $X\subseteq\mathfrak{B}\preccurlyeq_\mathcal{L}\mathfrak{A}$") ensures that we have a nice condensation phenomenon in the $M_\mathcal{L}^r$s, at least once $r$ is "strong enough" to code the relevant basic information, so these are in my opinion reasonably natural modifications of $L$ to consider.

Intuitively, due to the complexity of $i$ we might need the real $r$ to "unpack" things appropriately; for example, if we take $\mathcal{L}$ to be the "Tarski-Vaughtification" of second-order logic (see e.g. here) and use the obvious $i$ then we can really only hope for good behavior if we're given a real coding which second-order sentences are actually in $\mathcal{L}$. For this reason, I'm going to focus on the behavior of these models on a cone:

Definition: The ideal inner model theory of a good logic $\mathcal{L}$ is the first-order theory $$\mathsf{IIMT}_\mathcal{L}:=\{\varphi\in\mathsf{FOL}[\{\in\}]: \exists r\forall s\ge_Tr M_\mathcal{L}^s\models\varphi\}.$$

Due to the on-a-cone focus, the specific "coding map" $i$ used will play no role.

A secondary question focuses on combinatorics; here, I'm going to ask more specifically about large cardinals.

Question: Is it consistent (relative to large cardinals) that there is a good logic $\mathcal{L}$ such that $\mathsf{IIMT}_\mathcal{L}$ contains "There is a measurable cardinal?"

Consider the sublogic $\mathsf{Stv}$ of full second-order logic $\mathsf{SOL}$ consisting of all formulas $\varphi$ such that, whenever $\mathfrak{A}$ is a substructure of $\mathfrak{B}$ such that $\varphi^\mathfrak{B}\not=\emptyset$, we get $\varphi^\mathfrak{B}\cap\mathfrak{A}^{arity(\varphi)}\not=\emptyset$. (See this earlier question of mine for more detail; there I called this fragment "$\mathsf{SOL}_{TV}$.") $\mathsf{Stv}$ is significantly stronger than first-order logic (for example, $\mathsf{Stv}$ can define well-foundedness), but at the same time it has the full downward Lowenheim-Skolem property and so is also much weaker than second-order logic.

I'm curious about an analogue of $L$ built using $\mathsf{Stv}$ instead of first-order logic:

• $X_\alpha=L_\alpha$ for $\alpha\le\omega$.

• $X_{\omega+1}=L_{\omega+1}\cup\{s\}$, where $s$ is the set of (codes for) second-order formulas in finite languages which are elements of $\mathsf{Stv}$.

• For $\alpha>\omega+1$, we let $X_{\alpha+1}$ be the set of subsets of $X_\alpha$ which are first-order definable in the expansion of $X_\alpha$ by a binary relation symbol saying which $\mathsf{Stv}$-formulas are true in which finite-language structures in $X_\alpha$. (Hopefully it's clear what's going on here - the point is that $X_{\alpha+1}$ knows, for example, the $\mathsf{Stv}$-theory of $\mathcal{M}$ for each finite-language structure $\mathcal{M}\in X_\alpha$. I will of course add more detail if desired.)

• For $\lambda$ limit, we set $X_\lambda=\bigcup_{\alpha<\lambda}X_\alpha$.

The $L$-analogue I'm interested in is the class $X:=\bigcup_{\alpha\in\mathsf{Ord}}X_\alpha$. Since $\mathsf{Stv}$ satisfies the full downward Lowenheim-Skolem property, the $X$-hierarchy satisfies a version of condensation (basically, we have to look at the $X_\alpha$s equipped with the appropriate fragments of the $\models_{\mathsf{Stv}}$-relation) and so in particular $X\models\mathsf{GCH}$. At the same time, at least descriptive set theoretically $X$ is quite large (it has all the projective reals).

The obvious way to show that $X\models$ "There are no measurable cardinals" would be to show that, if $j:V\rightarrow M$ is the usual ultrapower embedding gotten from a measurable cardinal, then $j$ in fact respects $\mathsf{Stv}$ so that the restriction $j\upharpoonright X: X\rightarrow X$ would also be (first-order-)elementary. However, this isn't obvious to me given that measurables can live in $HOD$, the relevance being that Scott/Myhill showed that we get $HOD$ if we do anything like the above with full $\mathsf{SOL}$ (see the introduction of Kennedy/Magidor/Vaanaanen's Inner models for extended logics 1, and note that unless I'm missing something this paper and its follow-up don't solve the question here).

More generally, I'm interested in variations of the $L$-hierarchy built using any strong fragment of $\mathsf{SOL}$ with the downward Lowenheim-Skolem property (for example, Farmer S. proved that the absolute fragment can have dLS, and that fragment might actually be more natural here).

(The original version of this question was much narrower and less natural; but see the edit history if interested.)

Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ and having the finite use property and the strong downward Lowenheim-Skolem property together with, for each finite language $\Sigma$, an injection $i_\Sigma$ from the set of sentences $\mathcal{L}[\Sigma]$ to $\omega$. Letting $i=\{i_\Sigma:\Sigma$ a finite language$\}$, we can define an inner model $$M_\mathcal{L}^r:=L[\{\langle\mathfrak{A},n\rangle: \mathfrak{A}\models_\mathcal{L}i_{\mathit{Lang}(\mathfrak{A})}^{-1}(n)\},r]$$ for each real $r$. Strong dLS (= "For every $X\subseteq\mathfrak{A}$ in a finite language there is a $\mathfrak{B}$ with $\vert\mathfrak{B}\vert\le X\cdot\aleph_0$ and $X\subseteq\mathfrak{B}\preccurlyeq_\mathcal{L}\mathfrak{A}$") ensures that we have a nice condensation phenomenon in the $M_\mathcal{L}^r$s, at least once $r$ is "strong enough" to code the relevant basic information, so these are in my opinion reasonably natural modifications of $L$ to consider.

Intuitively, due to the complexity of $i$ we might need the real $r$ to "unpack" things appropriately; for example, if we take $\mathcal{L}$ to be the "Tarski-Vaughtification" of second-order logic (see e.g. here) and use the obvious $i$ then we can really only hope for good behavior if we're given a real coding which second-order sentences are actually in $\mathcal{L}$. For this reason, I'm going to focus on the behavior of these models on a cone:

Definition: The ideal inner model theory of a good logic $\mathcal{L}$ is the first-order theory $$\mathsf{IIMT}_\mathcal{L}:=\{\varphi\in\mathsf{FOL}[\{\in\}]: \exists r\forall s\ge_Tr M_\mathcal{L}^s\models\varphi\}.$$

Due to the on-a-cone focus, the specific "coding map" $i$ used will play no role.

A secondary question focuses on combinatorics; here, I'm going to ask more specifically about large cardinals.

Question: Is it consistent (relative to large cardinals) that there is a good logic $\mathcal{L}$ such that $\mathsf{IIMT}_\mathcal{L}$ contains "There is a measurable cardinal?"