2021/08/22 by Samson Leung, Leung, Samson
Computer Science · Economics, Econometrics and Finance · Mathematics · #03C48 (Primary) 03C45 #03C55 (Secondary) #Advanced Topology and Set Theory #FOS: Mathematics #Game Theory and Voting Systems #Logic (math.LO) #Logic, Reasoning, and Knowledge
paper · pdf · doi:10.48550/arxiv.2108.09708
openalex publication_date 2021/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For any abstract elementary class (AEC) \bf K with λ=LS(\bf K), the following holds: 1. K has an axiomatization in L(2λ)+,λ+, allowing game quantification. If \bf K has arbitrarily large models, the λ-amalgamation property and is categorical both in λ and λ+, then it has an axiomatization in Lλ+,λ+ with game quantification. These extend Kueker's result which assumes finite character and λ=ℵ0. 2. If K is universal and categorical in λ, then it is axiomatizable in Lλ+,λ+. 3. Shelah's celebrated presentation theorem asserts that for any AEC \bf K there is a first-order theory in an expansion of L(\bf K), and a set Γ of 2λ many T-types such that K=PC(T,Γ,L(\bf K)). We provide a better bound on |Γ| in terms of I2(λ,\bf K). 4. We present additional applications which extend, simplify and generalize results of Shelah and Shelah-Vasey. Some of our main results generalize to μ-AECs.