Submission date: 23 October 2023

Abstract:

There are many results in the literature where superstablity-like independence notions, without any categoricity assumptions, have been used to show the existence of larger models. In this paper we show that stability is enough to construct larger models for small cardinals assuming a mild locality condition for Galois types.

Theorem. Suppose λ<2^{ℵ_0}. Let K be an abstract elementary class with λ ≥ LS(K). Assume K has amalgamation in λ, no maximal model in λ, and is stable in λ. If K is (<λ^+, λ)-local, then K has a model of cardinality λ^{++}.

The set theoretic assumption that λ<2^{ℵ_0} and model theoretic assumption of stability in λ can be weaken to the model theoretic assumptions that |S^{na}(M)|< 2^{ℵ_0} for every M ∈ K_λ and stability for λ-algebraic types in λ. We further use the result just mentioned to provide a positive answer to Grossberg's question for small cardinals assuming a mild locality condition for Galois types and without any stability assumptions. This last result relies on an unproven claim of Shelah (Fact 4.5 of this paper) which we were unable to verify.

Mathematics Subject Classification: Primary: 03C48. Secondary: 03C45, 03C52, 03C55

Keywords and phrases:

Full text arXiv 2310.14474: pdf, ps.