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Preprint Number 1104

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1104. Sebastien Vasey
Shelah's eventual categoricity conjecture in universal classes. Part I
E-mail: sebv at cmu dot edu

Submission date:

Abstract:

We prove:
Theorem
Let K be a universal class. If K is categorical in cardinals of arbitrarily high cofinality, then K is categorical on a tail of cardinals.

The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes to abstract elementary classes (AECs) that satisfy a locality property and where certain prime models exist. Moreover assuming amalgamation we can give an explicit bound on the Hanf number and get rid of the cofinality restrictions:
Theorem
Let K be an AEC with amalgamation. Assume that K is fully LS(K)-tame and short and has primes over sets of the form M ∪ {a}. Write H2:=ℶ(2ℶ(2LS(K))+)+. If K is categorical in a λ > H2, then K is categorical in all λ' ≥ H2.

Mathematics Subject Classification: 03C48 (Primary), 03C45, 03C52, 03C55 (Secondary)

Keywords and phrases: Abstract elementary classes; Categoricity; Amalgamation; Forking; Independence; Classification theory; Superstability; Universal classes; Intersection property; Prime models.

Full text arXiv 1506.07024: pdf, ps.


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