Submission date: 2 February 2021

Abstract:

We generalize the Hart-Shelah example to higher infinitary logics. We build, for each natural number k greater than or equal to 2 and for each infinite cardinal lambda, a sentence ψ_k^λ of the logic L_{(2^λ)^+,ω} that (modulo mild set theoretical hypotheses around lambda 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)=(Mod(ψ_k^λ),≺_{(2^λ)^+,ω}) in the finite interval of cardinals λ,λ^+,...,λ^{+k}.

Mathematics Subject Classification: 03C48, 03C35, 03C75

Keywords and phrases: Categoricity, Infinitary Logic, Abstract Elementary Classes

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