The Hart-Shelah example, in stronger logics

2004/04/14 by Saharon Shelah, Shelah, Saharon, Andrés Villaveces +1
Computer Science · Mathematics · #03C35 #03C48 #03C75 #Advanced Algebra and Logic #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.math/0404258

openalex publication_date 2004/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We generalize the Hart-Shelah example \citeHaSh:323 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 \beth_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 \mathcal K^*(λ,k)=(Mod(ψ_k^λ),\prec_(2^λ)⁺,ω) in the finite interval of cardinals λ,λ⁺,…,λ^+k.

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