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

Published online by Cambridge University Press:  12 March 2014

Rami Grossberg*
Affiliation:
Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem, Israel
*
Current address of R. Grossberg: Department of Mathematics, Rutgers University, New Brunswick, N.J. 08903

Abstract

Let κ and λ be infinite cardinals such that λλ (we have new information for the case when κλ). Let T be a theory in Lκ +, ω of cardinality at most κ, let . Now define

Our main concept in this paper is is a theory in Lκ +, ω of cardinality κ at most, and φ(x, y) ϵ Lκ +, ω }. This concept is interesting because of

Theorem 1. Let T ⊆ Lκ +, ω of cardinality ≤ κ, and . If

then (∀χ > κ)I(χ, T) = 2 χ (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 . Here we continue that work by proving

Theorem 2. .

Theorem 3. For every κλ we have .

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 T ⊆ Lκ +, ω, and any set of formulas Lκ +, ω such that T ⊇ Lκ +, ω, if T is (, μ)-unstable for μ satisfying μμ*(λ,κ) = μ then T is -unstable (i.e. for every χλ, T is (, 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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1986

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Footnotes

1

We would like to thank the United States-Israel Binational Science Foundation for partially supporting this research. We are grateful to the referee for a very long and detailed referee report. The first author received support from Harvey Friedman, whom he thanks.

References

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