Fri Mar 18, 2016
60 Evans Hall, 4:10–6 PM
Guram Bezhanishvili (New Mexico State University)
The Algebra of Topology: Tarski’s Program 70 Years Later
In a slogan, Tarski’s program can be described as “creating an algebraic apparatus adequate for the treatment of portions of point-set topology” (McKinsey-Tarski, 1944). The “algebraic apparatus” can be developed in several ways. For example, we can look at the algebra of open sets of a topological space X or the powerset algebra of X equipped with topological closure (or interior). Both of these approaches are not only closely related, but are also widely used in logic. The first approach yields a topological representation of Heyting algebras, and hence provides topological completeness of intuitionistic logic (one of the first completeness results for intuitionistic logic, established by Tarski in the 1930s). On the other hand, the second approach provides a topological representation of closure algebras, thus yielding topological completeness of Lewis’s modal system S4. McKinsey and Tarski observed a close connection between these two approaches, which allowed them to prove that Gödel’s translation of intuitionistic logic into S4 is full and faithful. I will review this line of research of Tarski and his collaborators, discuss further advances in this direction, obtained in the second half of the twentieth century, and finish with the latest new developments.