The Löwenheim number

Intuition: The Löwenheim number is a measure of the downwards Löwenheim-Skolem property for logics

The Löwenheim number: The Löwenheim number of a logic $\mathcal{L}$, denoted $\ell (\mathcal{L})$, is the smallest cardinal $\mu$ such that if any arbitrary sentence $\phi$ of $\mathcal{L}$ has a model, the sentence has a model no larger than $\kappa$ (i.e., if $\mathcal{M}\models \phi ,$ then $|M|\le \mu$)

Formally, Let ${\kappa }_{\phi }$ denote the smallest cardinality of a model of $\phi ,$ otherwise if $\phi$ doesn’t have a model than ${\kappa }_{\phi }=0$. Then the The Löwenheim number is defined as the following $\ell (\mathcal{L})=$sup{${\kappa }_{\phi }|\phi$ is sentence in $\mathcal{L}$} If $\mathcal{L}$ doesn’t have a Löwenheim number, then we way $\ell (\mathcal{L})=\infty$

We can generalize this definition to a set of sentence $\le \kappa$

The Löwenheim-Skolem number: ${\ell }_{\kappa }(\mathcal{L})$ is the least cardinal $\mu$ such that any satisfiable set of $\mathcal{L}$-sentences has a model of power $\le \kappa$ has a model of power $\le \mu$ provided that such a cardinal $\mu$ exists, otherwise ${\ell }_{\kappa }(\mathcal{L})=\infty$.

Let ${\lambda }_{\Phi }$ denote the smallest cardinality of a model of $\Phi$, where $\Phi$ is a set of sentences of power $\le \kappa$. Then the Löwenheim-Skolem number is defined as follows: ${\ell }_{\kappa }(\mathcal{L})=$sup{${\lambda }_{\Phi }|\Phi$ is a set of sentences of power $\le \kappa$} $\to L(\mathcal{L})$ sup${}_{\kappa \in CARD}{l}_{\kappa }(\mathcal{L})$ is defined as the supremum of all cardinals ${\lambda }_{\Phi }$ such that an arbitrary set of $\mathcal{L}$-sentences, $\Phi$, has a model of power $\le$ $L(\mathcal{L})$. Equivalent to being the supremum of ${\ell }_{\kappa }(\mathcal{L})$ for any $\kappa$.

The Löwenheim number is the case where $\kappa =1$ (a single sentence rather than sets of sentences), i.e., $\ell (\mathcal{L})={\ell }_1(\mathcal{L})$

Note: ‘Cardinality’ refers to the size of a single domain, while power refers to the size of the overall structure, that is the sum of the cardinality of each domain. i.e., they differentiate in the case of many-sorted structures

The Löwenheim-Skolem-Tarski number: The Löwenheim-Skolem-Tarski number of the logic $\mathcal{L}$ is the smallest cardinal such that if $\mathcal{M}$ is a $\tau$-structure, then there is an elementary substructure of $\mathcal{M}$ of power $\le \kappa$. $\to$ This requires that the logic has a suitable notion of ‘elementary substructure’ (and elementary extension).

Denoted as $LST(\mathcal{L})$

Note: $\ell (\mathcal{L})\le {\ell }_{\kappa }(\mathcal{L})\le LST(\mathcal{L})$ always

E.g.,

• For $\mathcal{L}$ is first-order logic in a countable language, then $\ell (\mathcal{L})={\ell }_{\kappa }(\mathcal{L})=LST(\mathcal{L})={\aleph }_0$ This is a result of the Löwenheim-Skolem Theorem

Relation to Hanf numbers: $\ell (\mathcal{L})\le {\ell }_{\kappa }(\mathcal{L})\le LST(\mathcal{L})\le H(\mathcal{L})$; $\ell (\mathcal{L})\le h(\mathcal{L})\le h_{\kappa }(\mathcal{L})\le H(\mathcal{L})$; ${\ell }_{\kappa }\le h_{\kappa }(\mathcal{L})\le H(\mathcal{L})$

References

The Löwenheim-Skolem Theorem The Hanf Number Model Theoretic Logics.pdf Löwenheim number