Logic Seminar: A two-cardinal Ramsey operator on ideals

Time: Thursday, October 6, 11:00 am-12:00 noon

Place: Smith 120

Speaker:  Philip White, GWU

Title: A two-cardinal Ramsey operator on ideals

Abstract: Let I be a k-complete ideal. Similar to the one-cardinal ineffability operator of Baumgartner, Feng defined a one-cardinal Ramsey operator on I. A basic result of Feng is applying the one cardinal Ramsey operator to I yields a normal ideal. Feng also showed under what conditions the ideal given by applying the Ramsey operator is equivalently generated by a “pre-Ramsey” ideal as well as the Pi^1_(n+1) indescribability ideal.  Finally, Feng showed iterated use of the one-cardinal Ramsey operator forms a proper hierarchy. Feng was able to show these results for k+ iterations of the one-cardinal Ramsey operator by utilizing canonical functions. Similar to other results of Brent Cody and the presenter, these results in the one-cardinal setting can be generalized to a two-cardinal setting. The results of Feng will be discussed in detail as well as the analogous two-cardinal theorems of Brent Cody and the presenter.