A Two-Cardinal Ramsey Operator on Ideals

Abstract: Let $I$ be a $\kappa$-complete ideal on $\kappa$. 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 $< \kappa+$ 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 theorems of Feng will be discussed in detail as well as the analogous two-cardinal versions of Brent Cody and the presenter.

logic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic