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

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.

• Magidor has results on this. Jul 19, 2015 at 3:49
• Why did this get a -1? Seems a fine question to me. Jul 19, 2015 at 4:23
• Thomas, in your definition of the Löwenheim-Skolem number, shouldn't it be $\kappa\leq\text{max}(\kappa,|T|)$, rather than $<$? Jul 19, 2015 at 12:20
• Strange. Wikipedia defines it as I suggest, and that definition makes sense to me: en.wikipedia.org/wiki/L%C3%B6wenheim_number#Extensions. Jul 20, 2015 at 1:10
• @JoelDavidHamkins is right. You can't reasonably expect to get models of $T$ that are smaller than $|T|$. Jul 20, 2015 at 15:10

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.

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.