Categoricity Transference

Categoricity in Infinitary Logic If $T$ is categorical in $\lambda$ for $\lambda >h(T)$ then it is categorical in arbitrarily larger cardinals (larger than the Hanf number).

• If $h(\mathcal{L})=\infty$ (i.e., no weak upwards Löwenheim-Skolem property) then arbitrary categoricity is not guaranteed from any ‘point’ (cardinal). E.g., $T$ may be $\lambda$-categorical for any infinite cardinal $\lambda$, but not categorical in any other cardinal. This may also occur for $\lambda <h^{\prime }(\mathcal{L})=\kappa$ for $h^{\prime }(\mathcal{L})$ denotes the Hanf number for the property of categoricity.

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Background Information

Shelah’s presentation theorem: If $\mathcal{K}$ is an AEC in a vocabulary $\tau$ w/ $|\tau |\le LS(\mathcal{K})$, then there is a vocabulary $\sigma \supseteq \tau$ w/ cardinality $|LS(\mathcal{K})|$, a first-order (${\mathcal{L}}_{\omega ,\omega }$) theory $T$ of vocabulary $\sigma$, and a set $\Gamma$ of at most $2^{LS(\mathcal{K})}$ partial types such that: $\mathcal{K}=\{\mathcal{M}\upharpoonright \tau |\mathcal{M}\models T\text{ and }\mathcal{M}\text{ omits }\Gamma \}$

Categoricity Transference: Categoricity transference is a property where categoricity in a certain cardinal transfers (‘implies’) downwards or upwards to another cardinal (this is achieved via the Hanf number for categoricity).

Notation: $P(\mathcal{K},\lambda )$ denotes a property $P$ that applies to an AEC $\mathcal{K}$ in cardinality $\lambda$

Theorem: For a fixed set of classes $\mathcal{K}$ of a given kind (e.g., the classes of models defined by the infinitary logic ${\mathcal{L}}_{\mu ,v}$ for some fixed $\mu ,v$ of a given vocabulary). For any ‘property’ $P(\mathcal{K},\lambda )$ there is a cardinal $\kappa$ such that is $P(\mathcal{K},\lambda )$ holds for some $\lambda >\kappa$, then $P(\mathcal{K},\lambda )$ holds for arbitrarily large $\lambda$. Proof: Let ${\mu }_{\mathcal{K}}=$ sup$\{\lambda |P(\mathcal{K},\lambda )\text{ if such a max exists}\}$, then $\kappa =$ sup${\mu }_{\mathcal{K}}$ as $\mathcal{K}$ ranges over the set of all classes of a given type

Since there are a proper-class of sentences in ${\mathcal{L}}_{\infty ,\omega }$ (that ${\mathcal{L}}_{\infty ,\omega }[\tau ]$ forms a proper-class for an arbitrary vocabulary $\tau$), there are a proper-class of AEC with a given vocabulary.

Notation: For any AEC $\mathcal{K}$, let ${\kappa }_{\mathcal{K}}=$sup$\{|{\tau }_{\mathcal{K}}|,LS(\mathcal{K})\}$, where ${\tau }_{\mathcal{K}}$ denotes the vocabulary (or ‘similarity type’) of the AEC $\mathcal{K}$, and $LS(\mathcal{K})$ denotes the Löwenheim number of $\mathcal{K}$.

For any cardinal $\kappa$, there is only set-many AEC $\mathcal{K}$ with ${\kappa }_{\mathcal{K}}=\kappa$ since the AEC is determined via its restriction to model below the Löwenheim number (given a fixed vocabulary $\tau$).

A property $P$ is downwards closed if there is a ${\kappa }_0$ such that if $P(\mathcal{K},\lambda )$ holds with $\lambda >k_0$, then $P(\mathcal{K},\mu )$ holds if $k_0<\mu \le \lambda$.

Theorem: If a property $P$ is downwards closed, then for any $\kappa$ there is a cardinal $\mu$ for any AEC $\mathcal{K}$ w/ ${\mu }_{\kappa }=\kappa$, if some model $\mathcal{M}\in \mathcal{K}$ w/ property $P$ has cardinality greater than $\mu$ ($|M|>\mu$), then there is a model w/ property $P$ in all cardinals $\lambda >\mu$. $\to$ i.e., if an AEC is downwards closed for a property $P$, there is a Hanf number for $P$

Hanf Number for Properties: The Hanf number for a property $P$ for an AEC w/ ${\kappa }_{\mathcal{K}}=\kappa$ is $\mu$ if there is a model $\mathcal{M}\in \mathcal{K}$ w/ cardinality $>\mu$ that has property $P$, then there is a model $\mathcal{N}\in \mathcal{K}$ in all cardinalities greater than $\mu$. $\to$ Hanf’s argument can be applied via replacing ${\kappa }_{\mathcal{K}}$ w/ $|\tau |$, i.e., $H(|\tau |)$ instead of $H({\kappa }_{\mathcal{K}})$

We may then consider the property of categoricity, that is, every model of cardinality $|M|$ is isomorphic to $\mathcal{M}$, and so consider a Hanf number for categoricity (see below).

Hanf Number Construction: i) A first-order type $p$ is a consistent set of first-order formulae in a fixed number of finite variables. We say that $p$ is a complete type over a set $A$ if every formula w/ parameters from $A$ (or it’s negation) is in $p$ ii) Let $\eta (\lambda ,\kappa )$ denote the least cardinal $\kappa$ such that if a first-order theory $T$ w/ $|T|=\lambda$ has models of every cardinality $<\mu$ which omit each of a set $\Gamma$ of types. We call $\eta (\lambda ,\kappa )$ the Hanf function for omitting $\kappa$ types for theories of size $\lambda$ It is often written $\eta (\kappa )$ in the case of $\eta (\kappa ,\kappa )$ iii) We write $H(\kappa )$ for ${\beth }_{(2^{\kappa }{)}^{+}}$ iv) For a vocabulary $\tau ,H(\tau )$ means $H(|\tau |)$. W/ a fixed $\mathcal{K}$, we write $H_1$ for $H({\kappa }_{\mathcal{K}})=H(\text{sup}\{{\tau }_{\mathcal{K}},LS(\mathcal{K})\})$

Corollaries:

• $\eta (\kappa ,2^{\kappa })\le H(\kappa )={\beth }_{(2^{\kappa }{)}^{+}}$
• If $\mathcal{K}$ is an AEC and there is some $\mathcal{M}\in \mathcal{K}$ such that $|M|\ge H_1=H({\kappa }_{\mathcal{K}})$, then $\mathcal{K}$ has arbitrarily large models (i.e., There is a $\mathcal{N}\in \mathcal{K}$ such that for all $\lambda >H_1=H({\kappa }_{\mathcal{K}})$, then $|N|=\lambda$)

When ${\kappa }_{\mathcal{K}}=\text{max}\{LS(\mathcal{K}),|\tau (\mathcal{K})|\}$, then $H_1=H({\kappa }_{\mathcal{K}})$ is often called the Hanf number of $\mathcal{K}$. This is further equal to ${\beth }_{(2^{LS(\mathcal{K})}{)}^{+}}$ assuming $|{\tau }_{\mathcal{K}}|\le LS(\mathcal{K})$ (or ${\beth }_{(2^{|{\tau }_{\mathcal{K}}|}{)}^{+}}$ for $LS(\mathcal{K})\le |{\tau }_{\mathcal{K}}|$) Note: This is a bit misleading as a single AEC $\mathcal{K}$ cannot have a Hanf number, (a Hanf number is a maximum of all sentences and for all vocabularies of a given cardinality, as well as a fixed Löwenheim number for AEC’s). Rather the Hanf number for all AEC w/ the same ${\kappa }_{\mathcal{K}}$. For any model $\mathcal{M}$ w/ $|M|\ge \eta (\tau )$ where $|\tau |=LS(\mathcal{K})$, there are models of all cardinalities in the class that omit all types omitted in $\mathcal{M}$.

Notes:

• If the AEC is not fixed, then the Hanf number is a function of the (cardinality of the) vocabulary, $H_1=H(|{\tau }_{\mathcal{K}}|)$.
• $\tau (\mathcal{K})={\tau }_{\mathcal{K}}$, use is different in the specific context
• $H(\kappa )={\beth }_{(2^{\kappa }{)}^{+}}$ is an upper bound for the Hanf number (given it exists). Thus, it is possible for the Hanf number to smaller than ${\beth }_{(2^{\kappa }{)}^{+}}$, but not larger. $\to$ i.e., Considering first-order logic (${\mathcal{L}}_{\omega ,\omega }$) w/ in a countable language, $LS(\mathcal{K})={\aleph }_0$ and so the upper bound is $H_1={\beth }_{(2^{{\aleph }_0}{)}^{+}}$, even though by Morley’s categoricity theorem it is ${\aleph }_0$. By Shelah’s generalization, the Hanf number for first-order logic in an uncountable language of size $\kappa$, is $\kappa$. The reason for the use of Beth numbers is because it has been proven by Barwise that the Hanf number of any abstract logic corresponds to some Beth ${\beth }_n$, and the Hanf number for infinitary logics is the Beth of the successor.

Hanf numbers as described here: The Hanf Number, is the Hanf number for the property of the existence of models.

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Categoricity Transference

A larger Hanf number $H_2$ can be defined as follows: $H_2=H(H_1)=H(H({\kappa }_{\mathcal{K}}))={\beth }_{(2^{H_1}{)}^{+}}={\beth }_{(2^{H({\kappa }_{\mathcal{K}})}{)}^{+}}$ if $\mathcal{K}$ is fixed. Naturally this $⟹$ that we build a sequence of successively larger Hanf numbers where $H_1=H({\kappa }_{\mathcal{K}})$ and $H_{n+1}=H(H_n)$

The Hanf number $H_1$ is the Hanf number for arbitrarily large models (this ‘depends’ on the size of the vocabulary $|{\tau }_{\mathcal{K}}|$ and the Löwenheim Number $LS(\mathcal{K})$). While the Hanf number $H_2$ is used for tameness and locality assumptions, this can be used to show categoricity transference. i.e., For an AEC $\mathcal{K}$ that satisfies the amalgamation property (AP) and joint embedding property (JEP) and is categorical in $\lambda$ & ${\lambda }^{+}$, it can be shown that for some $\lambda \ge H_2$ then it is categorical in all $\mu \in [H_2,{\lambda }^{+}]$. Assuming $\mathcal{K}$ is $(\lambda ,\infty )$-tame for some $\chi <H_1$ allows one to extend this conclusion to all categoricity in all $\mu \ge H_2$.

We apply the Hanf function again to yield $H_2$, the Hanf number for categoricity transference because below $H_1$, models of arbitrary cardinalities is not guaranteed.

Downwards Categoricity Transference: Similar to how categoricity can be ‘transferred’ upwards from $\lambda$ to a successor cardinal ${\lambda }^{+}$ which can yield categoricity for an AEC $\mathcal{K}$ in arbitrarily large cardinals., categoricity can also be transferred downwards.

Theorem 14.14: Suppose $\mathcal{K}$ is categorical in some ${\lambda }^{+}$ w/ $\lambda \ge H_2$ and either i) $\mathcal{K}$ is $\lambda$-categorical; or ii) $\mathcal{K}$ admits $\lambda$-saturated unions then, i) $\mathcal{K}$ is categorical in every $\theta$ w/ $H_2\le \theta \le {\lambda }^{+}$; ii) If $\mathcal{K}$ is also $(H_2,\infty )$-weakly tame $\mathcal{K}$ is categorical in every $\theta$ w/ $H_2\le \theta$.

Theorem 15.11: Suppose $LS(\mathcal{K})<\lambda$ and that $\mathcal{K}$ has arbitrarily large models, the amalgamation property, and joint embedding property. Suppose $\mathcal{K}$ is ${\lambda }^{+}$-categorical, then i) $\mathcal{K}$ is categorical in all cardinals $H_2\le \mu \le {\lambda }^{+}$ ii) If $\mathcal{K}$ is $(\chi ,\infty )$-tame for some $\chi <{\lambda }^{+}$, then $\mathcal{K}$ is $\mu$-categorical for all $\mu \ge {\lambda }^{+}$ Where $\chi =H_1={\beth }_{(2^{\kappa }{)}^{+}}$

Conclusion 15.13: There is a cardinal $\mu$ depending on $\kappa$ such that if $\mathcal{K}$ is an AEC ${\kappa }_{\mathcal{K}}=\kappa$, and $\mathcal{K}$ is categorical in some successor cardinal ${\lambda }^{+}>\mu$, then $\mathcal{K}$ is categorical in all cardinal greater than $\mu$.

Proof: There are only set-many AEC $\mathcal{K}$ w/ ${\kappa }_{\mathcal{K}}$. Let ${\mu }_{\mathcal{K}}=\text{ sup}\{{\lambda }^{+}|\mathcal{K}\text{ is }{\lambda }^{+}\text{-categorical}\}$ given that such a supremum exists. Then let $\mu$ be the maximum of $H_2(=H_2(\kappa ))$ and $\text{sup}\{{\mu }_{\mathcal{K}}|{\kappa }_{\mathcal{K}}=\kappa \}$. If ${\kappa }_{\mathcal{K}}=\kappa$ and $\mathcal{K}$ is categorical in some successor cardinal greater than $\mu$, then $\mathcal{K}$ is categorical in arbitrarily large successor cardinals and in all cardinals greater than $H_2$ via theorem 15.11.

Notes:

• The smallest possible lower bound for an arbitrary infinitary sentence to be categorical is ${\beth }_{{\omega }_1}.$ For any $\alpha <{\aleph }_1$ there is a sentence ${\psi }_{\alpha }$ of ${\mathcal{L}}_{{\omega }_1,\omega }$ w/ no models in cardinals greater than ${\beth }_{\alpha }$ and w/ many models in each cardinal below ${\beth }_{\alpha }$. If $\psi$ asserts that the language contains only equality, then ${\psi }_{\alpha }\vee \psi$ is $\lambda$-categorical for $\lambda \in ({\beth }_{\alpha },\infty )$ (this property fails the amalgamation and joint embedding properties).
• Since $H_1<H_2$. If $H_1=\infty$ (i.e., there isn’t a point where arbitrarily large models are guaranteed), then $H_2=\infty$ (i.e., there isn’t a point where categoricity isn’t transferred arbitrarily upwards) and vice versa as $H_2=H(H_1)={\beth }_{(2^{H_1}{)}^{+}}=\infty$.

References

Categoricity Abstract Elementary Classes Stability Theory The Hanf Number Classification Theory AEClec.pdf