Reverse Mathematics and Pi^1_2 Comprehension

Stephen G. Simpson
Department of Mathematics
Pennsylvania State University

June 16, 2005

Note: This is an abstract of an invited one-hour talk, to be given in
a Recursion Theory workshop, part of the program Computational
Prospects of Infinity, June 20 to August 15, 2005, at the Institute
for Mathematical Sciences, National University of Singapore.


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.  An MF space is defined
to be a topological space of the form MF(P) with topology generated by
{ N_p | p in P }.  Here P is a poset, MF(P) is the set of maximal
filters on P, and N_p = { F in MF(P) | p in 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
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.