### On the Number of Nonisomorphic Models of an Infinitary Theory Which has the Infinitary Order Property. Part A

Rami Grossberg and Saharon Shelah
Source: J. Symbolic Logic Volume 51, Issue 2 (1986), 302-322.

#### Abstract

Let $\kappa$ and $\lambda$ be infinite cardinals such that $\kappa \leq \lambda$ (we have new information for the case when $\kappa < \lambda$). Let $T$ be a theory in $L_{\kappa^+,\omega}$ of cardinality at most $\kappa$, let $\varphi(\bar{x}, \bar{y}) \in L_{\lambda^+,\omega}$. Now define $\mu^\ast_\varphi (\lambda, T) = \operatorname{Min} \{\mu^\ast:$ If $T$ satisfies $(\forall\mu < \mu^\ast)(\exists M_\mu \models T)(\exists \{\bar{a}_i: i < \mu\} \subseteq M_\mu)// (\forall i, j < \mu)\lbrack i < j \Leftrightarrow M_\mu \models \varphi \lbrack \bar{a}_i, a_j\rbrack\rbrack \\\text{then} (\exists \varphi \in L_{\kappa^+,\omega})(\forall \chi > \kappa)(\exists M_\chi \models T)(\exists \{ a_i: i < \chi\} \subseteq |M_\chi|) (\forall i,j < \chi)\lbrack i < \chi) \lbrack i < j \Leftrightarrow M_\chi \models \varphi \lbrack a_i, a_j\rbrack\rbrack\}.$ Our main concept in this paper is $\mu^\ast_\varphi (\lambda, \kappa) = \operatorname{Sup}\{\mu^\ast(\lambda, T): T$ is a theory in $L_{\kappa^+,\omega}$ of cardinality $\kappa$ at most, and $\varphi (x, y) \in L_{\lambda^+,\omega}\}$. This concept is interesting because of THEOREM 1. Let $T \subseteq L_{\kappa^+,\omega}$ of cardinality $\leq \kappa$, and $\varphi (\bar{x}, \bar{y}) \in L_{\lambda^+,\omega}$. If $(\forall\mu < \mu^\ast (\lambda, \kappa))(\exists M_\mu \models T)(\exists\{\bar{a}_i: i < \mu\})(\forall i,j < \mu)\lbrack i < j \Leftrightarrow M_\mu \models \varphi \lbrack \bar{a}_i, \bar{a}_j\rbrack\rbrack$ then $(\forall_\chi > \kappa) I(\chi, T) = 2^\chi$ (where $I(\chi, T)$ stands for the number of isomorphism types of models of $T$ of cardinality $\chi$). Many years ago the second author proved that $\mu^\ast (\lambda, \kappa) \leq \beth_{(2^\lambda)^+}$. Here we continue that work by proving. THEOREM 2. $\mu^\ast (\lambda, \aleph_0) = \beth_{\lambda^+}$. THEOREM 3. For every $\kappa \leq \lambda$ we have $\mu^\ast (\lambda, \kappa) \leq \beth)_{(\lambda^\kappa)}^+$. For some $\kappa$ or $\lambda$ we have better bounds than in Theorem 3, and this is proved via a new two cardinal theorem. THEOREM 4. For every $\kappa \leq \lambda, T \subseteq L_{\kappa^+,\omega}$, and any set of formulas $\Lambda \subseteq L_{\lambda^+,\omega}$ such that $\Lambda \subseteq L_{\kappa^+,\omega}$, if $T$ is $(\Lambda,\mu)$-unstable for $\mu$ satisfying $\mu^{\mu^\ast(\lambda, \kappa)} = \mu$ then $T$ is $\Lambda$-unstable (i.e. for every $\chi \geq \lambda, T$ is $(\Lambda, \chi)$-unstable). Moreover, $T$ is $L_{\kappa^+,\omega}$-unstable. In the second part of the paper, we show that always in the applications it is possible to replace the function $I(\chi, T)$ by the function $IE(\chi, T)$, and we give an application of the theorems to Boolean powers.

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