The number of pairwise non-elementary-embeddable models

Journal of Symbolic Logic 54 (4):1431-1455 (1989)
We get consistency results on I(λ, T 1 , 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 $T \subseteq T_1$ are first order countable, T complete stable but ℵ 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(ℵ 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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