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We use continuous model theory to obtain several results concerning isomorphisms and embeddings between    II  1    factors and their ultrapowers. Among other things, we show that for any    II  1    factor   M  , there are continuum many nonisomorphic separable    II  1    factors that have an ultrapower isomorphic to an ultrapower of   M  . We also give a poor man's resolution of the Connes Embedding Problem: there exists a separable    II  1    factor such that all    II  1    factors embed into one of its ultrapowers.

Model theory of operator algebras III: elementary equivalence and II 1 factors