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Co-stationarity of the ground model
by
Natasha Dobrinen
Kurt Gödel Research Center for Mathematical Logic, University of Vienna
Coauthors: Sy-David Friedman
Given V, W models of ZFC with the same ordinals such that W contains V, and k a regular cardinal in W, let Pk(l) denote the collection of subsets of l of size less than k in W. We say that the ground model is co-stationary if Pk(l) \V is stationary in Pk(l). Gitik showed the following: Suppose k is a regular cardinal in W, and l is greater than or equal to (k+)W. If there is a real in W \V, then Pk(l) \V is stationary in Pk(l).
We consider problems of generalizing Gitik's Theorem to forcing extensions in which no reals are added. Assuming the existence of an w1-Erdös cardinal, we construct a model W of ZFC such that for each partial ordering P in W which adds a new subset of w1 and is (w3, w3, < w2)-distributive, Pw2(l) \V is stationary in Pw2(l) in the forcing extension WP for all l > w2. By a covering theorem of Magidor, this cannot be the case over a model V whose core model K has no w1-Erdös cardinal, for any P in V which adds no new sequences of length w. If there is a class of w1-Erdös cardinals, then we can construct a model of ZFC in which a generalization of Gitik's Theorem holds for partial orderings which add a new subset of w1 and are w2-c.c. We discuss these and related results for higher cardinals.
Date received: March 7, 2006
Copyright © 2006 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # casb-05.