The Topology of Model Theory
A departmental seminar talk, November 16 & 23, 2012.
Abstract: Model theory is a foundational subject. Like set theory and category theory, it provides a common language for talking about what different mathematicians do. Conversely, other parts of mathematics can illuminate features of model theory. This talk aims to present model theory in both ways: as foundational for mathematics, and as illuminated by other parts of mathematics.
One aim is then to present material that other model theorists should be able to assume in their own talks. This includes the notion of Morley rank, which can be seen as a special case of the topological notion of Cantor–Bendixson rank.
Another aim is to show how the compactness theorem of logic is really that a certain topological space is compact. As a consequence of the compactness theorem, this topological space is the full Stone space of a certain Boolean algebra (and is thus an example of the spectrum of a ring). Some model theory texts seem to suggest the converse, that the compactness theorem is a consequence of the compactness of Stone spaces; but this is incorrect.
My revised notes (23 pp., A5 paper, dated March 18, 2013):