ABSTRACT. The relation of semantic entailment, i.e. of a conclusion’s being true on every model of its premises, currently plays a central role in logic, and is arguably the canonical entailment-relation in most contexts. But it wasn’t always this way; the relation doesn’t come into its own until shortly before its starring role in the completeness theorem for first-order logic. This talk investigates the development of the notion of model from the mid-19th century to the early 20th, and the parallel emergence of logic’s concern with the relation of semantic entailment. We will be especially interested in clarifying some ways in which the emergence of the modern conceptions of model and of entailment are tied to a changing view of the nature of axiomatic foundations.

ABSTRACT. This is a talk in the branch of logic known as model theory, more precisely, in o-minimality. The first example of an o-minimal structure is the ordered algebraic structure on the set of real numbers and I shall focus on expansions of this structure. Being o-minimal means that the first order definable sets in the structure do not exhibit wild phenomena (this will be made precise). After discussing some basic theory of such structures I shall present some recent applications to diophantine geometry.