Axiomatizing AECs and applications

For any abstract elementary class (AEC) K with λ=LS(K), the following holds: has an axiomatization in L(2λ)+,λ+, allowing game quantification. If arbitrarily large models, λ-amalgamation property and is categorical both λ λ+, then it Lλ+,λ+ These extend Kueker's [10] result which assumes finite character λ=ℵ0. universal λ, axiomatizable Lλ+,λ+. Shelah's celebrated presentation theorem asserts that for AEC there a first-order theory expansion of L(K), set Γ 2λ many T-types such K=PC(T,Γ,L(K)). We provide better bound on |Γ| terms I2(λ,K). present additional applications extend, simplify generalize results Shelah [13], [15] Shelah-Vasey [17]. Some our main to μ-AECs.