# Sh:1092

• Baldwin, J. T., & Shelah, S. Hanf numbers for extendibility and related phenomena. Preprint.
• Abstract:
In this paper we discuss two theorems whose proofs depend on extensions of the Fraissé method. We prove the Hanf number for the existence of an extendible model (has a proper extension in the class. Here, this means an \infty,\omega-elementary extension) of a (complete) sentence of L_{\omega_1, \omega} is (modulo some mild set theoretic hypotheses that we expect to remove in a later paper) the first measurable cardinal. And we outline the description on an explicit L_{\omega_1, \omega}-sentence \phi_n characterizing \aleph_n for each n. We provide some context for these developments as outlined in the lectures at IPM.