# On the number of nonisomorphic models of an infinitary theory which has the infinitary order property. Part a

Journal of Symbolic Logic 51 (2):302-322 (1986)
 Abstract Let κ and λ be infinite cardinals such that κ ≤ λ (we have new information for the case when $\kappa ). Let T be a theory in L κ +, ω of cardinality at most κ, let φ(x̄, ȳ) ∈ L λ +, ω . Now define$\mu^\ast_\varphi (\lambda, T) = \operatorname{Min} \{\mu^\ast:$If T satisfies$(\forall\mu \kappa)(\exists M_\chi \models T)(\exists \{a_i: i Our main concept in this paper is $\mu^\ast_\varphi (\lambda, \kappa) = \operatorname{Sup}\{\mu^\ast(\lambda, T): T$ is a theory in L κ +, ω of cardinality κ at most, and φ (x, y) ∈ L λ +, ω }. This concept is interesting because of THEOREM 1. Let $T \subseteq L_{\kappa^+,\omega}$ of cardinality ≤ κ, and φ (x̄, ȳ) ∈ L λ +, ω . If $(\forall\mu then$(\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 μ μ * (λ, κ) = μ then T is Λ-unstable (i.e. for every χ ≥ λ, T is (Λ, χ)-unstable). Moreover, T is L κ +, 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 Keywords No keywords specified (fix it) Categories Logic and Philosophy of Logic (categorize this paper) DOI 10.2307/2274053 Options Save to my reading list Follow the author(s) My bibliography Export citation Edit this record Mark as duplicate Request removal from index
Our Archive

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 26,721

Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
References found in this work BETA

No references found.

Citations of this work BETA
Morasses, Square and Forcing Axioms.Charles Morgan - 1996 - Annals of Pure and Applied Logic 80 (2):139-163.
Similar books and articles

2009-01-28

167 ( #27,085 of 2,158,808 )