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.
Best Answer
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.