On Absoluteness of Categoricity in Abstract Elementary Classes

Shelah has shown in [4] that א1-categoricity for Abstract Elementary Classes (AEC’s) is not absolute in the following sense: There is an example K of an AEC (which is actually axiomatizable in the logic L(Q)) such that if 2א0 < 2א1 (the weak CH holds) then K has the maximum possible number of models of size א1, whereas if Martin’s Axiom at א1 (denoted by MAא1) holds then K is א1-categorical. In this note we extract the properties from Shelah’s example which make both parts work resulting in our definitions of condition A and condition B, and then we show that for any AEC satisfying these two conditions, neither of these implications can be reversed. 1. The model theoretic context In Shelah’s paper [4], the notion of Abstract Elementary Classes (AEC) was introduced, the idea being to write down basic properties of the first order elementary substructure relation. Definition 1. Let K be a class of models of a given similarity type and let ≺ be a partial ordering on K refining the ordinary substructure relation. The pair K = (K,≺) is an AEC, if (1) both K and ≺ are closed under isomorphism. (2) A ≺ C, B ≺ C and A ⊂ B imply A ≺ B (3) for any continuous ≺-chain (Aα)α<λ, (a) A = ⋃ α<λ Aα ∈ K (b) for all α < λ, Aα ≺ A (c) if Aα ≺ B for some B and all α < λ, then A ≺ B 2000 Mathematics Subject Classification. 03C35, 03C52, 03E35.