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Logic Seminar

Tuesday, February 9, 2016
3:00pm to 4:00pm
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Revisiting the first-order theories of McDuff's II_1 factors
Isaac Goldbring, Assistant Professor, Mathematics, University of Illinois at Chicago,

McDuff was the first to provide a family of continuum many pairwise-nonisomorphic separable II_1 factors. In a recent preprint, Boutonnet, Chifan, and Ioana proved that any ultrapowers of any two distinct McDuff examples are also nonisomorphic.  As a result, this shows that McDuff's examples are also pairwise non-elementarily equivalent, thus settling the question of how many first-order theories of II_1 factors there are.  From the model-theoretic point of view, this resolution of the question is not satisfying as we do not see an explicit family of sentences that distinguish the McDuff examples.  In this talk, I will present a partial resolution to this problem by discussing the following result:  If $M_\alpha$ and $M_\beta$ are two of McDuff's examples, where $\alpha,\beta \in 2^{\omega}$ are such that $\alpha|k=\beta|k$ but $\alpha(k)\not=\beta(k)$, then there must exist a formula of quantifier-complexity at most $6k+3$ on which they disagree.  The proof uses Ehrenfeucht-Fraisse games.  The talk represents joint work with Bradd Hart.

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