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The amalgamation spectrum

Published online by Cambridge University Press:  12 March 2014

John T. Baldwin
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, Il 60607, USA, E-mail: jbaldwin@uic.edu
Alexei Kolesnikov
Affiliation:
Towson University, Department of Mathematics, Towson, Md 21252, USA, E-mail: AKolesnikov@towson.edu
Saharon Shelah
Affiliation:
The Hebrew University of JerusalemJerusalem 91904, Israel Department of Mathematics, Rutgers University, New Brunswick, Nj 08854, USA, E-mail: shlhetal@math.huji.ac.il

Abstract

We study when classes can have the disjoint amalgamation property for a proper initial segment of cardinals.

For every natural number k, there is a class Kk, defined by a sentence in Lω1,ω that has no models of cardinality greater than ℶk + 1, but Kk has the disjoint amalgamation property on models of cardinality less than or equal to ℵk − 3 and has models of cardinality ℵk − 1.

More strongly, we can have disjoint amalgamation up to ℵ for < ω1, but have a bound on size of models.

For every countable ordinal , there is a class K defined by a sentence in Lω1,ω that has no models of cardinality greater than ℶω1, but K does have the disjoint amalgamation property on models of cardinality less than or equal to .

Finally we show that we can extend the to ℶ in the second theorem consistently with ZFC and while having ℵi ≪ ℶi for 0 < i < . Similar results hold for arbitrary ordinals with ∣∣ = k and Lk + ω.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2009

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