### Hanf numbers for extendibility and related phenomena

by Baldwin and Shelah. [BlSh:1092]

In this paper we discuss two theorems whose proofs depend on
extensions of the Fraisse 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.

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