TITLE: What sort of cardinal number is the Löwenheim–Skolem number for second-order logic?
QUESTION [18 upvotes]: 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.

REPLY [2 votes]: 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.