Tameness from Large Cardinal Axioms

TL;DR

This paper proves Shelah's Eventual Categoricity Conjecture from the existence of many strongly compact cardinals, establishing a link between large cardinal axioms and model-theoretic categoricity transfer.

Contribution

It demonstrates the first proof of the conjecture's consistency using large cardinal assumptions and introduces the concept of type shortness as a dual property to tameness.

Findings

Proves the conjecture assuming class many strongly compact cardinals.

Shows that AECs below a strongly compact cardinal are tame.

Establishes similar results for measurable and weakly compact cardinals.

Abstract

We show that Shelah's Eventual Categoricity Conjecture follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with $LS(K)$ below a strongly compact cardinal $\kappa$ is $< \kappa$ tame and applying the categoricity transfer of Grossberg and VanDieren. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, called \emph{type shortness}, and show that it follows similarly from large cardinals.