Journal of Symbolic Logic

The amalgamation spectrum

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.

Article information

Source
J. Symbolic Logic Volume 74, Issue 3 (2009), 914-928.

Dates
First available: 16 June 2009

http://projecteuclid.org/euclid.jsl/1245158091

Digital Object Identifier
doi:10.2178/jsl/1245158091

Zentralblatt MATH identifier
05609396

Mathematical Reviews number (MathSciNet)
MR2548468

Citation

Baldwin, John T.; Kolesnikov, Alexei; Shelah, Saharon. The amalgamation spectrum. Journal of Symbolic Logic 74 (2009), no. 3, 914--928. doi:10.2178/jsl/1245158091. http://projecteuclid.org/euclid.jsl/1245158091.