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Journal Article
On the Number of Nonisomorphic Models of an Infinitary Theory Which has the Infinitary Order Property. Part A
Rami Grossberg and Saharon Shelah
The Journal of Symbolic Logic
Vol. 51, No. 2 (Jun., 1986), pp. 302-322
Published
by: Association for Symbolic Logic
DOI: 10.2307/2274053
https://www.jstor.org/stable/2274053
Page Count: 21
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Topics: Logical theorems, Mathematical theorems, Cardinality, First order theories, Mathematical functions, Model theory, Mathematical models, Infinitary logic, Rudin Keisler order
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Abstract
Let κ and λ be infinite cardinals such that κ ≤ λ (we have new information for the case when $\kappa < \lambda$ ). Let T be a theory in Lκ+,ω of cardinality at most κ, let φ(x̄, ȳ) ∈ Lλ+,ω. 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 Lκ+,ω of cardinality κ at most, and φ (x, y) ∈ Lλ+,ω}. This concept is interesting because of THEOREM 1. Let $T \subseteq L_{\kappa^+,\omega}$ of cardinality ≤ κ, and φ (x̄, ȳ) ∈ Lλ+,ω. 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(χ, T) stands for the number of isomorphism types of models of T of cardinality χ). 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 κ ≤ λ we have $\mu^\ast (\lambda, \kappa) \leq \beth)_{(\lambda^\kappa)}^+$ . For some κ or λ 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 (Λ,μ)-unstable for μ satisfying μμ*(λ, κ) = μ then T is Λ-unstable (i.e. for every χ ≥ λ, T is (Λ, χ)-unstable). Moreover, T is Lκ+,ω-unstable. In the second part of the paper, we show that always in the applications it is possible to replace the function I(χ, T) by the function IE(χ, T), and we give an application of the theorems to Boolean powers.
The Journal of Symbolic Logic
© 1986 Association for Symbolic Logic