On chains of relatively saturated submodels of a model without the order property

Journal of Symbolic Logic 56 (1):124-128 (1991)
Abstract
Let M be a given model with similarity type L = L(M), and let L' be any fragment of L |L(M)| +, ω of cardinality |L(M)|. We call $N \prec M L'$ -relatively saturated $\operatorname{iff}$ for every $B \subseteq N$ of cardinality less than | N | every L'-type over B which is realized in M is realized in M is realized in N. We discuss the existence of such submodels. The following are corollaries of the existence theorems. (1) If M is of cardinality at least $\beth_{\omega_1}$ , and fails to have the ω order property, then there exists $N \prec M$ which is relatively saturated in M of cardinality $\beth_{\omega_1}$ . (2) Assume GCH. Let ψ ∈ L_{ω_1, ω, and let $L' \subseteq L_{\omega 1, \omega$ be a countable fragment containing ψ. If $\exists \chi > \aleph_0$ such that $I(\chi, \psi) , then for every $M \models \psi$ and every cardinal $\lambda of uncountable cofinality, M has an L'-relatively saturated submodel of cardinality λ
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