Journal of Symbolic Logic

The Number of Pairwise Non-Elementary-Embeddable Models

Saharon Shelah

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We get consistency results on $I(\lambda, T_1, T)$ under the assumption that $D(T)$ has cardinality $>|T|$. We get positive results and consistency results on $IE(\lambda, T_1, T)$. 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 $T \subseteq T_1$ are first order countable, $T$ complete stable but $\aleph_0$-unstable, $\lambda > \aleph_0$, and $|D(T)| > \aleph_0$, then $IE(\lambda, T_1, T) \geq \operatorname{Min}\{2^\lambda, \beth_2\}$. (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 $T \subseteq T_1, |T| = \aleph_0, |T_1| = \aleph_1, |D(T)| = \aleph_2$ and $IE(\aleph_2, T_1, T) = 1$. This paper (specifically the ZFC results) is continued in the very interesting work of Baldwin on diversity classes [B1]. Some more advances can be found in the new version of [Sh300] (see Chapter III, mainly \S7); they confirm 0.1, 0.2 and 14(1), 14(2).

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J. Symbolic Logic, Volume 54, Issue 4 (1989), 1431-1455.

First available in Project Euclid: 6 July 2007

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Shelah, Saharon. The Number of Pairwise Non-Elementary-Embeddable Models. J. Symbolic Logic 54 (1989), no. 4, 1431--1455.

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