Hanf number of the first stability cardinal in AECs
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.