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Logic and Computation Seminar

Monday, October 20, 2008 - 3:30pm

Nate Ackerman

University of Pennsylvania

Location

University of Pennsylvania

4C8

One of the features which distinguishes first order logic from higher order logics is the absoluteness of the satisfaction relation. Specifically if we have two standard models of set theory V_0 and V_1 with V_0 \subset V_1 then for any first order theory T and model M (both in V_0) V_0 thinks (M models T) if and only if V_1 thinks (M models T). Unfortunately as we move from first order to second order (and higher) we lose absoluteness. Given a second order T and a model M such that V_0 thinks (M models T) we have no reason to believe that V_1 will also think (M models T). However, in a wide class of theories T which occur in practice (such as topological spaces, complete lattices, ect.) for all models M where V_0 thinks (M models T), even if V_1 doesn't think (M models T) there is a "canonical" extension M' of M such that V_1 thinks (M' models T). In this context we call M' the "relativization" of M (with respect to T). In this series of talks we will formally introduce this notion of relativization and we will show there is a higher order theory of Grothendieck Toposes such that (assuming the axiom of choice) every model always has relativizations (with respect to the theory). This talk is based on part of my research during this last year.