Many structures in mathematics are created by putting forward axioms: axioms for a group, for a field, for a vector space, etc. Then we then study objects that satisfy those axioms (also called models of those axioms). Those would be groups, fields, vector spaces, etc.

One can actually study the relation between axiomatic theories and their models, to get many results about what models are possible, what size they might have, etc.

A surprising application is to set theory. One might think that the axioms of set theory uniquely determine what a set is. But that is not so: there are many different, inequivalent models of set theory (assuming that set theory is consistent, i.e. that one cannot derive a contradiction from the axioms!).
This phenomenon was discovered with the solution of the Continuum Hypothesis, whose answer depends on which model of set theory is taken. Here, we say that the Continuum Hypothesis is undecidable: the axioms of set theory alone do not settle the question.

Come along at 7:30 in MS.05 as Sam Derbyshire introduces model theory and its applications to various other areas of mathematics. No prior knowledge is assumed, so students from all years should be able to understand.

As usual, after the talk we will head over to the Duck for some refreshments. Hope to see you there!