Are there logics that lack a Löwenheim number?

Question

This answer contains the definition of the Löwenheim number. The question is about second-order logic.

The Löwenheim number is the smallest cardinal $λ$ so that if a theory $T$ has a model then it has a model of size less than $\max(|T|,λ)$.

This can be paraphrased into the following. It says the same thing but makes the quantification over theories and models explicit. Let $\lambda$ be an ordinal and let $L$ be a logic. Let $\mathrm{num}(L)$ denote the Löwenheim number of $L$.

$$ \mathrm{num}(L) \le \lambda \iff \forall T \mathop. ( (\exists M. M \models T) \to (\exists M\models T \mathop. |M| < \max(|T|, \lambda)))$$

One thing I'm curious about, though, is why we're guaranteed the existence of any $\lambda$s satisyfing the above property for second-order logic. So, my question is twofold. Why does second-order logic have a Löwenheim number at all? Are there any logics, that are even more badly behaved than second-order logic, that lack a Löwenheim number?

Answer 1

There's an important subtlety here: we're working within a fixed signature (see e.g. Definition $3$ in this paper). If $\mathcal{L}$ is a logic such that $\mathcal{L}[\tau]$ (= the set of $\mathcal{L}$-sentences in the signature $\tau$) is a set for every set-sized signature $\tau$, then there are only set-many satisfiable $\mathcal{L}[\tau]$-theories and so there is a cardinal $\kappa$ such that every satisfiable $\mathcal{L}[\tau]$-theory $T$ has a model of size $<\max\{\kappa,\vert T\vert\}$.

(The situation is of course more interesting for a reasonably-nice-but-class-sized logic like $\mathcal{L}_{\infty,\omega}$, but let's ignore that for now.)

What if we try to look at all signatures at once? Well, then even first-order logic (indeed a tiny fragment thereof) fails to have such a cardinal! For an infinite cardinal $\kappa$, let $\tau_\kappa=\{c_i:i<\kappa\}$ be a large set of constant symbols and let $T_\kappa=\{c_i\not=c_j: i<j<\kappa\}$. Then $T_\kappa$ is satisfiable but has no model of size $<\max\{\kappa,\vert T_\kappa\vert\}$ (all of $T_\kappa$'s models have size $\ge\kappa$ and $\vert T_\kappa\vert=\kappa$).

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As an aside, note that there are modifications we could make to add some nuance to the situation. For example, what if we replace "$<\max\{\kappa,\vert T\vert\}$" with "$\le\max\{\kappa,\vert T\vert\}$" but still ask about all signatures at once (call this the "weird Lowenheim number")? Now downward Lowenheim-Skolem implies that $\mathsf{FOL}$ does have a weird Lowenheim number, namely $\omega$. However, $\mathsf{SOL}$ does not:

Given an infinite cardinal $\kappa$ let $\sigma_\kappa$ be the signature consisting of $\kappa$-many unary predicates, $\sigma_\kappa=\{U_i:i<\kappa\}$, together with one further unary predicate $D$. Write an $\mathsf{SOL}[\sigma_\kappa]$-theory $S_\kappa$ saying that each $U_i$ is infinite, $\vert U_i\vert< \vert U_j\vert$ whenever $i<j$, $U_i\subseteq D$ for each $i$, and $D$ has strictly smaller cardinality than the whole structure. Then $S_\kappa$ is satisfiable, $\vert S_\kappa\vert=\kappa$, but all of its models have cardinality $\ge\kappa^+$.