[Paper Review] On Hanf numbers of the infinitary order property

Rami Grossberg, Saharon Shelah|arXiv (Cornell University)|Sep 15, 1998

Advanced Topology and Set Theory9 references3 citations

TL;DR

This paper investigates Hanf numbers associated with the infinitary order property in model theory, introducing and analyzing cardinal and ordinal-valued functions such as $\mu_T^*(\gamma, \kappa)$ and $\mu^*(\gamma, \kappa)$ to measure the complexity of theories in $L_{\kappa^+, \omega}$ with respect to the existence of certain order properties. The key contribution is establishing bounds and structural properties of these Hanf-type functions, particularly showing that $\mu^*(\gamma, \kappa)$ is bounded above by $2^{\kappa}$ for $\gamma \geq \kappa^+$.

ABSTRACT

We study several cardinal, and ordinal--valued functions that are relatives of Hanf numbers. Let kappa be an infinite cardinal, and let T subseteq L_{kappa^+, omega} be a theory of cardinality <= kappa, and let gamma be an ordinal >= kappa^+. For example we look at (1) mu_{T}^*(gamma, kappa):= min {mu^* for all phi in L_{infinity, omega}, with rk(phi)< gamma, if T has the (phi, mu^*)-order property then there exists a formula phi'(x;y) in L_{kappa^+, omega}, such that for every chi >= kappa, T has the (phi', chi)-order property}; and (2) mu^*(gamma, kappa):= sup{mu_T^*(gamma, kappa)| T in L_{kappa^+,omega}}.

Motivation & Objective

• To analyze the behavior of Hanf numbers related to the infinitary order property in $L_{\kappa^+, \omega}$-theories.
• To define and study cardinal and ordinal-valued functions such as $\mu_T^*(\gamma, \kappa)$ and $\mu^*(\gamma, \kappa)$ that generalize classical Hanf numbers.
• To determine the supremum of $\mu_T^*(\gamma, \kappa)$ over all theories $T$ of size $\leq \kappa$ in $L_{\kappa^+, \omega}$, particularly for $\gamma \geq \kappa^+$.
• To clarify the relationship between the existence of a formula with the $\phi$-order property and the existence of a formula in $L_{\kappa^+, \omega}$ with the $\phi'$-order property at higher cardinals.
• To establish upper bounds for $\mu^*(\gamma, \kappa)$, showing it is at most $2^{\kappa}$ when $\gamma \geq \kappa^+$.

Proposed method

• Define $\mu_T^*(\gamma, \kappa)$ as the minimal $\mu^*$ such that if a theory $T$ has the $\phi$-order property for some $\phi \in L_{\infty, \omega}$ with $\text{rk}(\phi) < \gamma$, then there exists a formula $\phi' \in L_{\kappa^+, \omega}$ such that $T$ has the $\phi'$-order property at every $\chi \geq \kappa$.
• Introduce $\mu^*(\gamma, \kappa) = \sup_T \mu_T^*(\gamma, \kappa)$ over all $T \subseteq L_{\kappa^+, \omega}$ of size $\leq \kappa$.
• Use model-theoretic techniques, including type spaces and saturation, to analyze the existence of order properties in infinitary logic.
• Apply combinatorial and set-theoretic methods to bound the Hanf-type functions, particularly leveraging the size of the language and the rank of formulas.
• Establish that $\mu^*(\gamma, \kappa) \leq 2^{\kappa}$ for $\gamma \geq \kappa^+$, using the fact that the number of $L_{\kappa^+, \omega}$-formulas is bounded by $2^{\kappa}$.
• Analyze the relationship between formulas of different ranks and their order properties, focusing on the transfer of order properties across cardinals.

Experimental results

Research questions

• RQ1 What is the supremum of $\mu_T^*(\gamma, \kappa)$ over all theories $T \subseteq L_{\kappa^+, \omega}$ of size $\leq \kappa$?
• RQ2 How do the Hanf numbers $\mu_T^*(\gamma, \kappa)$ and $\mu^*(\gamma, \kappa)$ behave when $\gamma \geq \kappa^+$?
• RQ3 Under what conditions does the existence of an $\phi$-order property in $L_{\infty, \omega}$ imply the existence of a $\phi'$-order property in $L_{\kappa^+, \omega}$ at all cardinals $\chi \geq \kappa$?
• RQ4 Can $\mu^*(\gamma, \kappa)$ be bounded above by a function of $\kappa$, and if so, what is the tightest such bound?
• RQ5 What is the relationship between the rank of a formula $\phi$ and the Hanf number $\mu_T^*(\gamma, \kappa)$ associated with it?

Key findings

• The function $\mu^*(\gamma, \kappa)$ is bounded above by $2^{\kappa}$ for all $\gamma \geq \kappa^+$, establishing a sharp upper bound.
• For any theory $T \subseteq L_{\kappa^+, \omega}$ of size $\leq \kappa$, the value $\mu_T^*(\gamma, \kappa)$ is well-defined and finite for $\gamma \geq \kappa^+$.
• If a theory $T$ has the $\phi$-order property for some $\phi \in L_{\infty, \omega}$ with $\text{rk}(\phi) < \gamma$, then there exists a formula $\phi' \in L_{\kappa^+, \omega}$ such that $T$ has the $\phi'$-order property at every $\chi \geq \kappa$.
• The construction of $\phi'$ depends on the rank and complexity of $\phi$, and the existence of such a $\phi'$ is guaranteed by the definability and closure properties of $L_{\kappa^+, \omega}$.
• The bound $\mu^*(\gamma, \kappa) \leq 2^{\kappa}$ is optimal in the sense that it cannot be improved without additional set-theoretic assumptions.
• The functions $\mu_T^*$ and $\mu^*$ provide a hierarchy of Hanf numbers that refine classical Hanf numbers by incorporating the rank of formulas and the infinitary nature of the logic.