Reverse Mathematics and Pi^1_2 Comprehension

Stephen G. Simpson
Department of Mathematics
Pennsylvania State University
http://www.personal.psu.edu/t20/

January 17, 2007

Note: This is an abstract of an invited 50-minute talk, to be given
January 21, 2007 at the Massachusetts Institute of Technology, as part
of a conference in honor of Richard Shore's 60th birthday.

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 Pi^1_2 comprehension.  If P is a poset, let
MF(P) be the space of maximal filters on P.  Here MF(P) has the
obvious topology generated by basic open sets N_p = { F in MF(P) | p
in F }, p in P.  An MF space is defined to be a topological space of
the form MF(P).  If P is countable, we say that MF(P) is 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
regular and is homeomorphic to a countably based MF space.  We show
that the converse statement, "every regular, countably based MF space
is homeomorphic to a complete separable metric space," is equivalent
to Pi^1_2-CA_0.  The equivalence is proved in the weaker system
Pi^1_1-CA_0.  This is the first example of a theorem of core
mathematics which is provable in second-order arithmetic and implies
Pi^1_2 comprehension.