Axiomatizing AECs and applications

Abstract

For any abstract elementary class (AEC) ${\bf K}$ with $λ=LS({\bf K})$, the following holds: 1. $K$ has an axiomatization in $L_{(2^λ)^+,λ^+}$, 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 $L_{λ^{+},λ^{+}}$ with game quantification. These extend Kueker's result which assumes finite character and $λ=\aleph_0$. 2. If $K$ is universal and categorical in $λ$, then it is axiomatizable in $L_{λ^+,λ^+}$. 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 $2^λ$ many $T$-types such that $K=PC(T,Γ,L({\bf K}))$. We provide a better bound on $|Γ|$ in terms of $I_2(λ,{\bf K})$. 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.