The Hanf Number

Intuition: The Hanf number is a measure of (fragment of) the upwards Löwenheim-Skolem property of a logic

Pinning Down Cardinals: $\Phi \subseteq \mathcal{L}[\tau ]$ pins down the cardinal $\kappa$ if and only if $\Phi$ has a model of cardinality $\kappa ,$ but such that $\Phi$ does not have models of arbitrarily high cardinalities (i.e., does not have a model of size $\lambda \ge \kappa$).

Let $h_{\kappa }(\mathcal{L})$ be the supremum of all cardinals that can be pinned down by a set of $\mathcal{L}$-sentences of power $\kappa$.

More formally, Denote ${\lambda }_{\Phi }$ as the cardinals $\lambda$ pinned down by the set of $\mathcal{L}$-sentences $\Phi$. Then the Hanf number is defined as follows: $h_{\kappa }(\mathcal{L})=$sup{${\lambda }_{\Phi }|\Phi$ is a set of $\mathcal{L}$-sentences $\le \kappa$} $\to$ sup${}_{\kappa \in CARD}{h}_{\kappa }(\mathcal{L})$ is defined as the supremum of all cardinals ${\lambda }_{\Phi }$ pinned down by an arbitrary set of $\mathcal{L}$-sentences (equivalent to the supremum of all $h_k(\mathcal{L})$ for each $\kappa$). A majority of logic will not have such a number since increasing the size of the vocabulary $\tau$, via adding additional non-logical symbols, will usually increase $h_{\kappa }(\mathcal{L})$

The Hanf number of a logic $\mathcal{L}$, denoted $h(\mathcal{L})$, is the case where $\kappa =1$, i.e., $h(\mathcal{L})=h_1(\mathcal{L})$ That is, the supremum of all cardinals pinned down by an arbitrary $\mathcal{L}$-sentence $\phi$.

Let $h(\mathcal{L})=\mu ,$ if an arbitrary set of $\mathcal{L}$-sentences $\Phi$ has model $\mathcal{M}$ of power $\kappa \ge \mu$, the $\Phi$ has a model of size $\lambda$ for all $\lambda \ge \kappa$.

If $\mathcal{L}$ does not have a Hanf number, then $h(\mathcal{L})=\infty$

Per the definition of the Hanf number, if an arbitrary set of $\mathcal{L}$-sentences $\Phi$, of power $\le \kappa$, has a model $\mathfrak{A}\models \Phi$ such that $\mathfrak{A}\ge h_{\kappa }(\mathcal{L})$, then $\Phi$ has a models of arbitrarily size $\lambda \ge h_{\kappa }(\mathcal{L})$. So intuitively, the Hanf number is the point/cardinal at which the existence of a model of this size or greater then guarantees arbitrarily size (i.e., the point at which models start to have arbitrarily large size).

Note: $h(\mathcal{L})\le h_{\kappa }(\mathcal{L})$ always

Proposition: If $\Phi \subseteq \mathcal{L}[\tau ]$ is an arbitrary set of $\mathcal{L}$-sentences such that $|\Phi |\le \kappa$ and $\Phi$ has arbitrarily large models below $h_{\kappa }(\mathcal{L})$ (that is $\mathfrak{A}\models \Phi$ and such that $|\mathfrak{A}|=\lambda$ for every $\lambda <h_k(\mathcal{L})$), then $\Phi$ has arbitrarily large models, that is, $\Phi$ has a model of every cardinality. $\to$ i.e., The pattern doesn’t stop

If $h(\mathcal{L})=\infty$, then models of arbitrarily large cardinality are not guaranteed. An implication of this is the existence of maximally large (w/ respect to cardinality) models of an arbitrary set of $\mathcal{L}$-sentences, $\Phi$. This existence of maximal models already occurs if all models of $\Phi$ have power less than $h(\mathcal{L})=\mu$ (for $\mu$ is some cardinal), then it will have maximally sized models. In first-order logic, if a theory has no infinite models, then it will have some maximally large model which is finite. $h(\mathcal{L})>{\aleph }_0$ allows for maximal models which are infinite in size $\to$ This is equivalent to the logic $\mathcal{L}$ having no fragments of the upwards Löwenheim-Skolem property. $\to$The larger $h(\mathcal{L})$ is, the more cardinals $\mathcal{L}$ can characterize between. If $h(\mathcal{L})=\infty$, then $\mathcal{L}$ can characterize (pinning down) all cardinals. $⟹$ the logic $\mathcal{L}$ can differentiate between the cardinalities of it’s own models. i.e., in first-order logic, a consequence of the Löwenheim-Skolem theorem is that we can have uncountable models satisfying things such as “I am countable” (and vice versa), by pinning down more cardinals (and so increasing $h(\mathcal{L}))$), the more accurate the internal view of the models will match the external view w/ respect to its size.

Extension: A stronger form of Hanf number, $H_{\kappa }(\mathcal{L})$ can be defined which asserts that if an arbitrary set of sentences $\Phi$ of power $\le \kappa$ has a model $\mathfrak{A}$ such that $|\mathfrak{A}|\ge H_{\kappa }(\mathcal{L})$ than said model has an elementary extensions $\mathfrak{B}⪰\mathfrak{A}$ of arbitrarily large cardinality. $\to$ By said definition we derive $h_{\kappa }(\mathcal{L})\le H_{\kappa }(\mathcal{L})$

Hanf Number Results:

• $h({\mathcal{L}}_{{\kappa }^{+},\omega })<{\beth }_{(2^{\kappa }{)}^{+}}$
• The Hanf number of first-order logic in vocabulary $\tau$ of size $\kappa$ is max$\{\tau ,{\aleph }_0\}$

References

The Löwenheim-Skolem Theorem The Löwenheim Number Model Theoretic Logics.pdf Categoricity