Logic Seminar - Alexi Block Gorman

November 15, 2019
Tuesday, December 3, 2019 - 1:50pm to 2:50pm
Dulles Hall 012
Logic Seminar

Title: When does a subgroup of the real numbers embed nicely into a "generic" subgroup of the reals?

Speaker: Alexi Block Gorman - UIUC

Abstract: Model theorists have created examples and non-examples of interesting properties by thinking about "generic" sets, substructures, and reducts living in models whose theories are otherwise tame. Here, being generic means there is a witness from that set and from its complement for every formula that only uses existential quantifiers to define a "nice" (e.g. open and infinite) set. Intuitively, a set would only be called generic if both it and its complement intersect every infinite set in any arity of the ambient structure unless there is a really good reason it should not. In this talk, we will focus on expansions of o-minimal structures (for our purposes, an expansion of the reals in which all unary definable sets are a finite unions of points and intervals) and we will characterize when the theory of the reals expanded by a dense and codense divisible subgroup embeds nicely into a generic subgroup. The characterization is geometric in nature, and we can apply the characterization to both additive subgroups of the reals and multiplicative subgroups of the reals without zero. We will discuss an array of interesting examples and non-examples that highlight the versatility of the characterization.

 

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