Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-22T21:05:12.492Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[AlGr]Albert, M. and Grossberg, R., Rich models, this Journal, vol. 55 (1990), pp. 1292–1298.Google Scholar
[Gr]Grossberg, R., Indiscernible sequences in a model which fails to have the order property, this Journal, vol. 56 (1991), pp. 115–123.Google Scholar
[GrSh]Grossberg, R. and Shelah, S., The number of nonisomorphic models of an infinitary theory which has the order property. Part A, this Journal, vol. 51 (1966), pp. 302–322.Google Scholar
[Sh1]Shelah, S., Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1978.Google Scholar
[Sh2]Shelah, S., Universal classes. Part I, Classification theory (proceedings, Chicago, 1985; Baldwin, J. T., editor), Lecture Notes in Mathematics, vol. 1292, Springer-Verlag, Berlin, 1988, pp. 264–419.Google Scholar
[Sh3]Shelah, S., A combinatorial problem: stability and order for models and theories in infinitary languages, Pacific Journal of Mathematics, vol. 41 (1972), pp. 247–261.CrossRefGoogle Scholar