For a logic $\mathcal{L}$, let the compactness number of $\mathcal{L}$ (if it exists) be the least $\kappa$ such that every $(<\kappa)$-satisfiable $\mathcal{L}$-theory is satisfiable. Note that there is no restriction here on the cardinality of the language of the theory in question.

For example, an uncountable cardinal $\kappa$ is strongly compact iff it is the compactness number of its own infinitary logic $\mathcal{L}_{\kappa,\kappa}$; more interestingly, Magidor showed that $\mathsf{SOL}$ has a compactness number iff there is an extendible cardinal, in which case its compactness number is the least extendible cardinal.

My question is:

What is the strength of "For every $\kappa$, the compactness number of $\mathcal{L}_{\kappa,\kappa}$ exists?"

I'm a bit embarrassed to say how little I know about this. Obviously a proper class of strongly compact cardinals suffices, but in fact I already don't know that this isn't an outright theorem of $\mathsf{ZFC}$!

• A reasonable first step is to consider what happens in $\mathsf{ZFC+V=L}$. There is of course a single $\mathcal{L}_{\omega_1,\omega_1}$-sentence which characterizes exactly the (structures isomorphic to) levels of $L$, and so one might try to show that the existence of a compactness number for $\mathcal{L}_{\omega_1,\omega_1}$ lets us construct a model-theoretic phenomenon in the class of levels of $L$ which we know can't exist (see e.g. here). But I see no way to actually get that off the ground.

For a logic $\mathcal{L}$, let the compactness number of $\mathcal{L}$ (if it exists) be the least $\kappa$ such that every $(<\kappa)$-satisfiable $\mathcal{L}$-theory is satisfiable. Note that there is no restriction here on the cardinality of the language of the theory in question.

For example, an uncountable cardinal $\kappa$ is strongly compact iff it is the compactness number of its own infinitary logic $\mathcal{L}_{\kappa,\kappa}$; more interestingly, Magidor showed that $\mathsf{SOL}$ has a compactness number iff there is an extendible cardinal, in which case its compactness number is the least extendible cardinal.

My question is:

What is the strength of "For every $\kappa$, the compactness number of $\mathcal{L}_{\kappa,\kappa}$ exists?"

EDIT: Originally I said that I didn't know anything relevant, but I just noticed that one of the suggested related questions is very relevant, namely this one: there it is shown for example that the existence of a compactness number for $\mathcal{L}_{\omega_1,\omega_1}$ already implies the existence of a measurable cardinal, or more technically that the existence of a compactness number for $\mathcal{L}_{\omega_1,\omega_1}$ is equivalent to the existence of an $\omega_1$-strongly compact cardinal. A natural guess based on that is that the principle in question is equivalent to "For every $\kappa$ there is a $\kappa$-strongly compact cardinal," but I haven't had a chance to read through the argument in detail so I'm not too confident here.

For a logic $\mathcal{L}$, let the compactness number of $\mathcal{L}$ (if it exists) be the least $\kappa$ such that every $(<\kappa)$-satisfiable $\mathcal{L}$-theory is satisfiable. Note that there is no restriction here on the cardinality of the language of the theory in question.

For example, an uncountable cardinal $\kappa$ is strongly compact iff it is the compactness number of its own infinitary logic $\mathcal{L}_{\kappa,\kappa}$; more interestingly, Magidor showed that $\mathsf{SOL}$ has a compactness number iff there is an extendible cardinal, in which case its compactness number is the least extendible cardinal.

My question is:

What is the strength of "For every $\kappa$, the compactness number of $\mathcal{L}_{\kappa,\kappa}$ exists?"

I'm a bit embarrassed to say how little I know about this. Obviously a proper class of strongly compact cardinals suffices, but in fact I already don't know that this isn't an outright theorem of $\mathsf{ZFC}$!

• A reasonable first step is to consider what happens in $\mathsf{ZFC+V=L}$. There is of course a single $\mathcal{L}_{\omega_1,\omega_1}$-sentence which characterizes exactly the (structures isomorphic to) levels of $L$, and so one might try to show that the existence of a compactness number for $\mathcal{L}_{\omega_1,\omega_1}$ lets us construct a model-theoretic phenomenon in the class of levels of $L$ which we know can't exist (see e.g. here). But I see no way to actually get that off the ground.

For a logic $\mathcal{L}$, let the compactness number of $\mathcal{L}$ (if it exists) be the least $\kappa$ such that every $(<\kappa)$-satisfiable $\mathcal{L}$-theory is satisfiable. Note that there is no restriction here on the cardinality of the language of the theory in question.

For example, an uncountable cardinal $\kappa$ is strongly compact iff it is the compactness number of its own infinitary logic $\mathcal{L}_{\kappa,\kappa}$; more interestingly, Magidor showed that $\mathsf{SOL}$ has a compactness number iff there is an extendible cardinal, in which case its compactness number is the least extendible cardinal.

My question is:

What is the strength of "For every $\kappa$, the compactness number of $\mathcal{L}_{\kappa,\kappa}$ exists?"

I'm a bit embarrassed to say how little I know about this. Obviously a proper class of strongly compact cardinals suffices, but in fact I already don't know that this isn't an outright theorem of $\mathsf{ZFC}$!

• A reasonable first step is to consider what happens in $\mathsf{ZFC+V=L}$. There is of course a single $\mathcal{L}_{\omega_1,\omega_1}$-sentence which characterizes exactly the (structures isomorphic to) levels of $L$, and so one might try to show that the existence of a compactness number for $\mathcal{L}_{\omega_1,\omega_1}$ lets us construct a model-theoretic phenomenon in the class of levels of $L$ which we know can't exist (see e.g. here). But I see no way to actually get that off the ground.