Abstract
Theorem: For each 2 ≤ k < ω there is an \( L_{\omega _1 ,\omega } \)-sentence ϕk such that
(1) ϕk is categorical in μ if μ≤ℵk−2;
(2) ϕk is not ℵk−2-Galois stable
(3) ϕk is not categorical in any μ with μ>ℵk−2;
(4) ϕk has the disjoint amalgamation property
(5) For k > 2
(a) ϕk is (ℵ0, ℵk−3)-tame; indeed, syntactic first-order types determine Galois types over models of cardinality at most ℵk−3;
(b) ϕk is ℵm-Galois stable for m ≤ k − 3
(c) ϕk is not (ℵk−3, ℵk−2).
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The first author is partially supported by NSF grant DMS-0500841.
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Baldwin, J.T., Kolesnikov, A. Categoricity, amalgamation, and tameness. Isr. J. Math. 170, 411–443 (2009). https://doi.org/10.1007/s11856-009-0035-8
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DOI: https://doi.org/10.1007/s11856-009-0035-8