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 K_k defined by a sentence in L_ω₁,ω that has no models of cardinality greater than ℶ_k+1, but K_k has the disjoint amalgamation property on models of cardinality less than or equal to ℵ_k-3 and has models of cardinality ℵ_{k}-1.

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

Theorem B For every countable ordinal α, there is a class K_α defined by a sentence in L_ω₁,ω that has no models of cardinality greater than ℶ_ω₁, but K does have the disjoint amalgamation property on models of cardinality less than or equal to ℵ_α.

Finally we show that we can extend the ℵ_α to ℶ_α in the second theorem consistently with ZFC and while having ℵ_i≪ ℶ_i for 0< i≤ α. Similar results hold for arbitrary ordinals α with |α|=κ and L_κ⁺,ω.