Louise Hay Logic Seminar
Martin Zeman
University of California, Irvine
Inner models and absoluteness
Abstract: Absoluteness of statements between transitive models of set theory is a well-studied topic. The classical result in this area is the well-known Shoenfield absoluteness theorem stating that every \Sigma^1_2 statement is absolute in this sense. Absoluteness for statements of higher complexity requires large cardinals. By classical work of Martin and Solovay, the existence of proper class of measurable cardinals guarantee that \Sigma^1_3 statements are absolute for generic extensions. The well-known high-level result in the area is Woodin's theorem on the generic absoluteness of \Sigma^2_2 statements. There are other types of absoluteness results which concern models that are not generic extensions. Among the most well-known are results on \Sigma^1_3 correctness of the core model, where the classical version is due to Jensen and the modern version is due to Steel. In the talk we focus on the absoluteness of \Sigma^1_3 statements with respect to any extension, not just generic, which correctly computes the successor of at least one regular cardinal. Under the suitable large cardinal hypothesis we show that such an extension is \Sigma^1_3 absolute. This is a joint work with Andres Caicedo.
Thursday November 7, 2013 at 4:00 PM in SEO 427