Reverse Mathematics and Π_{2}^{1} Comprehension

Stephen G. Simpson Department of Mathematics Pennsylvania State University

            http://www.math.psu.edu/simpson/

November 13, 2006

Note: This is an abstract of an invited one-hour talk, to be given November 17, 2006, at the Logic Colloquium in the Department of Mathematics at the University of Florida.

Abstract

This is joint work with Carl Mummert. We initiate the reverse mathematics of general topology. We show that a certain metrization theorem is equivalent to Π_{2}^{1} comprehension. An MF space is defined to be a topological space of the form MF(P) with topology generated by {N_{p} | p ∈ P }. Here P is a poset, MF(P) is the set of maximal filters on P, and N_{p} = {F ∈ MF(P) | p ∈ F }. If the poset P is countable, the space MF(P) is said to be countably based. The class of countably based MF spaces can be defined and discussed within the subsystem ACA_{0} of second order arithmetic. One can prove within ACA_{0} that every complete separable metric space is homeomorphic to a countably based MF space which is regular. We show that the converse statement, “every countably based MF space which is regular is homeomorphic to a complete separable metric space,” is equivalent to Π_{2}^{1}-CA_{0}. The equivalence is proved in the weaker system Π_{1}^{1}-CA_{0}. This is the first example of a theorem of core mathematics which is provable in second order arithmetic and implies Π_{2}^{1} comprehension.