| Time: | 12 - 1:20 p.m. |
| Room: |
OSC 201
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| Speaker: |
Stephen Simpson Professor Department of Mathematics Pennsylvania State University |
| Title: |
Reverse mathematics and Π12 comprehension |
| 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 Π12 comprehension. An MF space is defined
to be a topological space of the form MF(P) with topology generated by
{ Np | p ε P }. Here P is a poset, MF(P) is the set of maximal
filters on P, and Np = { F ε MF(P) | p ε 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 ACA0 of second order arithmetic. One can prove within
ACA0 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
Π12 CA0. The equivalence is proved in the weaker system
Π11 CA0. This is the first example of a theorem of core
mathematics which is provable in second order arithmetic and implies
Π12 comprehension.
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| Organizer's note: | Lunch will be provided.
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