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

Title: | Regularity properties and forcing absoluteness |

Seminar: | Logic Seminar |

Speaker: | Daisuke Ikegami, University of California, Berkeley |

Abstract: |

Some properties of sets of reals such as Lebesgue measurability and the Baire property are called regularity properties. A set of reals with a regularity property can be approximated by some simple sets (e.g. closed sets, Borel sets) in terms of some smallness (e.g. Lebesgue null sets, meager sets) and many sets (e.g. Sigma^1_1 sets) enjoy regularity properties while there is a set of reals without such regularity properties using the Axiom of Choice.
Forcing absoluteness is the absoluteness of the statements between a ground model and its forcing extensions. Many simple statements such as Sigma^1_2 formulas are forcing absolute in this sense (Shoefield Absoluteness Theorem).
We discuss the complexity of sets of reals without such regularity properties and the complexity of statements which are not forcing absolute and the establish the general equivalence results between regularity properties and forcing absoluteness for a large class of tree-type forcings. A sample theorem is as follows:
Theorem
The following are equivalent:
(1) every Sigma^1_3 statement is absolute via Sacks forcing, and
(2) no Delta^1_2 sets of reals are Bernstein.
In this talk, we focus on examples of the general equivalence rather than stating and proving the general results. |

### Room Reservation Information

Room Number: | MB315 |

Date: | 04 / 24 / 2012 |

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