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Preprint Number 559

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559. Saharon Shelah
A.E.C. with not too many models
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Submission date: 21 February 2013.

Abstract:

Consider an a.e.c. (abstract elementary class), that is, a class K of models with a partial order refining inclusion (submodel) which satisfy the most basic properties of an elementary class. Our test question is trying to show that the function dot I (lambda, K), counting the number of models in K of cardinality lambda up to isomorphism, is “nice”, not chaotic, even without assuming it is sometimes 1, i.e. categorical in some lambda's. We prove here that for some closed unbounded class C of cardinals we have (a), (b) or (c) where
     (a) for every lambda in C of cofinality aleph_0, dot I (lambda, K) greater than or equal to lambda
     (b) for every lambda in C of cofinality aleph_0 and M belongs to K_lambda, for every cardinal kappa greater than or equal to lambda there is N_kappa of cardinality kappa extending M (in the sense of our a.e.c.)
     (c) mathfrak k is bounded; that is, dot I (lambda, K) = 0 for every lambda large enough (equivalently lambda greater than or equal to beth_{delta_*} where delta_* = (2^{LST(mathfrak k)})^+).

Recall that an important difference of non-elementary classes from the elementary case is the possibility of having models in K, even of large cardinality, which are maximal, or just failing clause (b).

Mathematics Subject Classification: 03C45, 03C48, 03C55, 03C75

Keywords and phrases:

Full text arXiv 1302.4841: pdf, ps.


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