Hanf number of the first stability cardinal in AECs

TL;DR

This paper establishes a lower bound for the Hanf number related to stability in AECs, explores variations of order properties, and proves the independence of joint embedding property necessity, using syntactic methods and Galois Morleyization.

Contribution

It provides a new lower bound for Hanf numbers in AECs, introduces variations of order properties, and proves the independence of joint embedding property necessity for type-counting lemmas.

Findings

Lower bound for Hanf numbers in AECs established.

Variations of order and syntactic order properties introduced.

Independence of joint embedding property necessity proved.

Abstract

We show that $\beth_{(2^{\operatorname{LS}({\bf K})})^+}$ is the lower bound to the Hanf numbers for the length of the order property and for stability in stable abstract elementary classes (AECs). Our examples satisfy the joint embedding property, no maximal model, $(<\aleph_0)$-tameness but not necessarily the amalgamation property. We also define variations on the order and syntactic order properties by allowing the index set to be linearly ordered rather than well-ordered. Combining with Shelah's stability theorem, we deduce that our examples can have the order property up to any $\mu<\beth_{(2^{\operatorname{LS}({\bf K})})^+}$. Boney conjectured that the need for joint embedding property for two type-counting lemmas is necessary. We solved the conjecture by showing it is independent of ZFC. Using Galois Morleyization, we give syntactic proofs to known stability results assuming a monster model.