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.

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