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 ψ k λ of the logic L ( 2 λ ) + , ω that (modulo mild set theoretical hypotheses around λ and assuming 2 λ λ + m ) is categorical in λ + , … , λ + k − 1 but not in ℶ k + 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 ) = ( M o d ( ψ k λ ) , ≺ ( 2 λ ) + , ω ) in the finite interval of cardinals λ , λ + , … , λ + k .