Gabriel Conant : Distance Structures for Generalized Metric Spaces

Posted by Kostyantyn Slutskyy , part of the Logic Seminar.

At

March 31, 2015, 3 p.m.

In

SEO 1227

Abstract

Suppose $M$ is a metric space taking distances in an arbitrary totally ordered commutative monoid $R$. When considered as a discrete first-order structure in a relational language, nonstandard models of the theory of $M$ can no longer be considered as metric spaces over $R$, in a way coherent with the first-order theory. To solve this problem, we construct a monoid extension $R^*$ of $R$, with the property that any model of the theory of $M$ is a metric space over $R^*$ under a "type-definable" metric. In the case that $R$ is countable, and $M$ is the countable Urysohn space over $R$, we use $R^*$ to characterize quantifier elimination for the theory of $M$.