The Hart-Shelah example, in stronger logics

Abstract

We generalize the Hart-Shelah example [10] to higher infinitary logics. We build, for each natural number k≥2 and for each infinite cardinal λ, a sentence ψkλ of the logic L(2λ)+,ω that (modulo mild set theoretical hypotheses around λ and assuming 2λ<λ+m) is categorical in λ+,…,λ+k−1 but not in ℶk+1(λ)+ (or beyond); we study the dimensional encoding of combinatorics involved in the construction of this sentence and study various model-theoretic properties of the resulting abstract elementary class K⁎(λ,k)=(Mod(ψkλ),≺(2λ)+,ω) in the finite interval of cardinals λ,λ+,…,λ+k.