A Set-Theoretic Approach to Model Theory
The City University of New York
Although one always employs logic in proofs, the foundations of many branches of mathematics appear to be predominantly set-theoretic: one defines a topological space to be a pair (X, τ) consisting of a set X and a collection τ of subsets satisfying certain well-known properties; one defines a group to be a pair (G, μ) consisting of a set G and a subset μ ⊂ G × G × G as the binary operation satisfying certain well-known properties (of course, for a group one needs a bit more to handle the identity); etc. There are advantages to this commonality, particularly if one is well-versed in category theory: one can move from one area to the other and still have a fairly good idea of what the major problems are and the sort of techniques one might expect to see. In contrast, in Model Theory, the foundation appears to be heavily based on logic, and as a result the language and terminology can seem foreign to those who work in more widely publicized areas of mathematics. Rather than “sets of groups”, one hears about “sets of formulas”; rather than products (Cartesian, fibered, direct, or semi-direct), one hears of “ultraproducts”; rather than “reducing to a simpler case”, one is told about “eliminating quantifiers”.
In this talk I will indicate how some of the basic ideas of Model Theory can be formulated set-theoretically, that is, in the topological and algebraic spirit indicated above.
Professor Churchill is on the faculty at Hunter College of CUNY and also a member of the doctoral faculty in the mathematics program at The CUNY Graduate Center.