On Absoluteness of Categoricity in Abstract Elementary Classes

Abstract

Shelah has shown that $\aleph_1$-categoricity for Abstract Elementary Classes (AECs) 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 $2^{\aleph_0}<2^{\aleph_1}$ (the weak CH holds) then $K$ has the maximum possible number of models of size $\aleph_1$, whereas if Martin's Axiom at $\aleph_1$ (denoted by MA) holds then $K$ is $\aleph_1$-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.