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We show that $\beth_{(2^{\operatorname{LS}({\bf K})})^+}$ is the lower boundto the Hanf numbers for the length of the order property and for stability instable abstract elementary classes (AECs). Our examples satisfy the jointembedding property, no maximal model, $(<\aleph_0)$-tameness but notnecessarily the amalgamation property. We also define variations on the orderand syntactic order properties by allowing the index set to be linearly orderedrather than well-ordered. Combining with Shelah's stability theorem, we deducethat our examples can have the order property up to any$\mu<\beth_{(2^{\operatorname{LS}({\bf K})})^+}$. Boney conjectured that theneed for joint embedding property for two type-counting lemmas is necessary. Wesolved the conjecture by showing it is independent of ZFC. Using GaloisMorleyization, we give syntactic proofs to known stability results assuming amonster model.

Hanf number of the first stability cardinal in AECs