Categoricity and infinitary logics

Abstract

We point out a gap in Shelah's proof of the following result: $\mathbf{Claim}$ Let $K$ be an abstract elementary class categorical in unboundedly many cardinals. Then there exists a cardinal $\lambda$ such that whenever $M, N \in K$ have size at least $\lambda$, $M \le N$ if and only if $M \preceq_{L_{\infty, \text{LS} (K)^+}} N$. The importance of the claim lies in the following theorem, implicit in Shelah's work: $\mathbf{Theorem}$ Assume the claim. Let $K$ be an abstract elementary class categorical in unboundedly many cardinals. Then the class of $\lambda$ such that: 1) $K$ is categorical in $\lambda$; 2) $K$ has amalgamation in $\lambda$; and 3) there is a good $\lambda$-frame with underlying class $K_\lambda$ is stationary. We give a proof and discuss some related questions.