Kurt Gödel Research Center

2004 seminar talk: CH and the saturation of the nonstationary ideal on omega_1

Talk held by Andres Caicedo (KGRC) at the KGRC seminar on 2004-03-23.

Abstract

The saturation of the nonstationary ideal on omega_1 was shown consistent (with ZFC) from a strong form of determinacy by Steel and VanWesep in the early 80's. Their techniques produced a model where CH fails. It has been an open question since then whether a model can be produced where the ideal is saturated and CH holds. Although this problem is still open, significant progress towards a (negative) solution was made by Woodin in the 90's. Specifically, Woodin proved that the saturation of the ideal contradicts CH, *in the presence of large cardinals*. In fact, a definable counterexample is produced. However, no such definable counterexample can exist if the large cardinals are absent from the picture, and apparently a completely new idea is necessary to settle the problem in this case. A nice side effect of Woodin's techniques is the development of the theory of P_max.

In this talk I plan to present Woodin's result, together with its limitations.