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

Published online by Cambridge University Press:  12 March 2014

Rami Grossberg
Rami Grossberg*
Affiliation:
Department of Mathematics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213
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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 ≺ ML′-relatively saturated iff for every B ⊆ N of cardinality less than ∥N∥ every L′-type over B which 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 ℶω1, and fails to have the ω order property, then there exists N ≺ M which is relatively saturated in M of cardinality ℶω1.

(2) Assume GCH. Let ψ ∈ Lω1, ω, and let L′ ⊆ Lω1, ω be a countable fragment containing ψ. If ∃χ > ℵ0 such that I(χ, ψ) < 2χ, then for every M ⊨ ψ and every cardinal λ < ∥M∥ of uncountable cofinality, M has an L′-relatively saturated submodel of cardinality λ.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 1 , March 1991 , pp. 124 - 128
Copyright
Copyright © Association for Symbolic Logic 1991

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References

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