What sort of cardinal number is the Löwenheim–Skolem number for second-order logic?

Question

In their paper “On Löwenheim–Skolem–Tarski numbers for extensions of first order logic”, Magidor and Väänänen make the following statement:

“For second order logic, $\mathrm{LS}(L^{2})$ [the Löwenheim–Skolem number for second order logic—my comment] is the supremum of $\Pi_{2}$-definable ordinals..., which means that it exceeds the first measurable, the first $\kappa^{+}$-supercompact $\kappa$, and the first huge cardinal if they exist”.

[“The Löwenheim–Skolem number $\mathrm{LS}(L^{2})$ of second order logic $L^{2}$ is the smallest cardinal $\kappa$ such that if a theory $T\subset L^{2}$ has a model, it has a model of cardinality $\le\max(\kappa,|T|)$”, and “$L^{2}$ extends first order logic with quantifiers of the form $\exists R\,\phi(R,x_0,\dots,x_{n-1})$, where the second order variable $R$ ranges over $n$-ary relations on the universe for some fixed $n$”—my comment also but substantially quoting the authors.]

Assume that the first measurable, the first $\kappa^{+}$-supercompact $\kappa$, and the first huge cardinals exist. What type of large cardinal, then, is $\mathrm{LS}(L^{2})$? If the answer is known, please provide the reference.

Answer 1

The following is due to Magidor:

Theorem 1. Is $\kappa$ is a strong cardinal, then $LS(L^2) < \kappa.$

The proof if easy. Let $T \subseteq L^2$ be a theory and let $A$ be a model for $T$. e may assume the universe is some cardinal $\delta.$ Take some cardinal $\beta > \beth_{\omega}(\delta)$, and let $j: V \to M$ witness $\kappa$ is $\beta$-strong. It is easily seen $M\models$``$A \models T$'', so $M \models \exists B( B \models T, |B| < j(\kappa))$. By elementarity in $V$, $T$ has a model of size $< \kappa.$

Also note that for any theory $T \subseteq L^2,$ there is a least $\delta_T$ such that if $T$ has a model, then it has a model of size $< \delta_T.$ Then $LS(L^2)=\sup \{\delta_T: T$ as above $\}$, so $LS(T^2)$ can be singular, even though it can be above some very large cardinals.

Answer 2

The Löwenheim number (compared to LS, this uses sentences rather than theories) for the second order logic $L^2$ is the least $κ$ such that $V_κ$ satisfies all true $Σ_2$ sentences. This $κ$ has cofinality $ω$, and $V_κ$ does not satisfy ZFC, but $κ$ is a limit of large cardinals. For example, if there is a proper class of huge cardinals, then $κ$ is a limit of huge cardinals, and same with other $Σ_2$ properties.

Assuming countable vocabulary, the Löwenheim-Skolem (LS) number for $L^2$ is the least $κ$ such that $V_κ$ satisfies all true $Σ_2$ sentences with real parameters. $ω_1≤\operatorname{cf}(κ)≤c$, where $c$ is the cardinality of the continuum. As before, $V_κ$ does not satisfy ZFC, but $κ$ is a limit of large cardinals.

If we allowed a proper class of constant symbols, then $L^2$ would not have a Löwenheim-Skolem number. Still, even then, if $V_κ ≺_{Σ_2} V$, then every theory of cardinality $<κ$ with an $L^2$ model has such a model of cardinality $<κ$. For reference, among $κ$ with $V_κ ≺_{Σ_2} V$, the following are in strictly increasing order of the least example (if any, if not assuming large cardinals): $\operatorname{cf}(κ)=ω$, $\operatorname{cf}(κ)=ω_1$, $V_κ⊨\text{ZFC}$, $\operatorname{cf}(κ)=ω_1 ∧ V_κ⊨\text{ZFC}$, $κ$ is regular, $κ$ is Mahlo, $κ$ is weakly compact, $κ$ is measurable, $κ$ is strong. Note that for every strong $κ$, $V_κ ≺_{Σ_2} V$.

Also, the Löwenheim-Skolem-Tarski (LST) number of a logic $L$ is the smallest cardinal $κ$ such that every structure for $L$ has an elementary substructure of size $≤κ$. As noted in the paper in the question, the LST number for $L^2$ (with countable vocabulary) is the least supercompact cardinal, or none if none exists; the Vopěnka's principle holds iff every logic with set-sized vocabulary has an LST number.