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