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Preprint Number 904

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904. Will Boney and Sebastien Vasey
Categoricity and infinitary logics
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Submission date: 13 August 2015.

Abstract:

We use model-theoretic forcing to show:

Theorem
Let K be an abstract elementary class categorical in unboundedly many cardinals. Then there exists a cardinal λ such that whenever M, N in K have size at least λ, M ≤ N if and only if M \preceq_{L_{∞, LS(K)^+}} N. This fixes a gap in Shelah's proof of the following result:

Theorem
Let K be an abstract elementary class categorical in unboundedly many cardinals. Then the class of λ such that:
1) K is categorical in λ;
2) K has amalgamation in λ; and
3) there is a good λ-frame with underlying class K_λ is stationary.

Mathematics Subject Classification: 03C48 (Primary), 03C25, 03C45, 03C52, 03C55, 03C75, 03B60, 18D99 (Secondary)

Keywords and phrases:

Full text arXiv 1508.03316: pdf, ps.


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