For more information about this meeting, contact Stephen Simpson, Jason Rute, Jan Reimann.

Title: | An introduction to reverse mathematics |

Seminar: | Logic Seminar |

Speaker: | Stephen G. Simpson, Penn State |

Abstract: |

Reverse mathematics is a program of research in the foundations of mathematics. A basic discovery, emphasized by Hilbert and Bernays in the 1930s, is that the proofs of almost all theorems of core mathematics (analysis, algebra, geometry, combinatorics, ...) are straightforwardly formalizable in subsystems of second-order arithmetic. Reverse mathematics is a series of case studies, initiated in the 1970s, in which specific core mathematical theorems are examined in order to determine the smallest subsystem of second-order arithmetic in which the given theorem is provable. The program of reverse mathematics is very rich, with many sub-programs and many open problems. Many concepts from mathematical logic come into play, including Turing computability, the Turing jump operator, basis theorems, relative hyperarithmeticity, the hyperjump, proof-theoretic ordinals, and nonstandard models of arithmetic. Over the past 40 years, some interesting conclusions have emerged. For instance, it turns out that many core mathematical theorems are logically equivalent to the axioms needed to prove them, and this leads to a robust classification of such theorems up to logical equivalence. Indeed, many such theorems fall into only five equivalence classes, the so-called "Big Five." An area of recent interest is the reverse mathematics of measure theory, which involves algorithmic randomness. |

### Room Reservation Information

Room Number: | MB315 |

Date: | 09 / 02 / 2014 |

Time: | 02:30pm - 03:45pm |