### The amalgamation spectrum

John T. Baldwin, Alexei Kolesnikov, and Saharon Shelah
Source: J. Symbolic Logic Volume 74, Issue 3 (2009), 914-928.

#### Abstract

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

Theorem A 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 ℵ{k}-1.

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

Theorem B For every countable ordinal α, there is a class Kα 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 ℵα.

Finally we show that we can extend the ℵα to ℶα in the second theorem consistently with ZFC and while having ℵi≪ ℶi for 0< i≤ α. Similar results hold for arbitrary ordinals α with |α|=κ and Lκ⁺,ω.

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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1245158091
Digital Object Identifier: doi:10.2178/jsl/1245158091
Zentralblatt MATH identifier: 05609396
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