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The number of pairwise non-elementarily-embeddable models

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah*
Institute of Mathematics, The Hebrew University, Jerusalem, Israel Department of Mathematics, Simon Fraser University, Burnaby, British Columbia B5A 1S6, Canada Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903


We get consistency results on I(λ, T, T) under the assumption that D(T) has cardinality > ∣T∣. We get positive results and consistency results on IE(λ, T 1, T).

The interest is model-theoretic, but the content is mostly set-theoretic: in Theorems 1–3, combinatorial; in Theorems 4–7 and 11(2), to prove consistency of counterexamples we concentrate on forcing arguments; and in Theorems 8–10 and 11(1), combinatorics for counterexamples; the rest are discussion and problems. In particular:

(A) By Theorems 1 and 2, if TT 1 are first order countable, T complete stable but ℵ0-unstable, λ > ℵ0, and ∣D(T)∣ > ℵ0, then IE(λ, T 1, T) > Min{2 λ , ℶ2}.

(B) By Theorems 4,5,6 of this paper, if e.g. V = L, then in some generic extension of V not collapsing cardinals, for some first order TT 1, ∣T∣ = ℵ0, ∣T 1∣ = ℵ1, ∣D(T)∣ = ℵ2 and IE(ℵ2, T1, T) = 1.

This paper (specifically the ZFC results) is continued in the very interesting work of Baldwin on diversity classes [Bl]. Some more advances can be found in the new version of [Sh300] (see Chapter III, mainly §7); they confirm 0.1, 0.2 and 14(1), 14(2).

Research Article
Copyright © Association for Symbolic Logic 1989

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This work was partially supported by grant number A1030 from the Natural Sciences and Engineering Research Council of Canada, and also by grants from the U.S. National Science Foundation and the U.S.-Israel Binational Science Foundation. This paper is number 262 in the list on pp. 398–418 of [Sh300].


I thank Rami Grossberg for many corrections, and in particular for rewriting the proof of Theorem 1; and also John Baldwin for adding explanations. The paper's volume has grown considerably under their pressure. I also thank Alice Leonhardt for the beautiful typing of the manuscript.



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