Department of Mathematics
Stationary subsets of ω1 and models of set
theory
Andres Caicedo
Universitaet Vien, Kurt Goedel Research Center
A set X has size ω1 if it is uncountable, and every infinite subset of X has either the size of the natural numbers, or else it has the size of X, i.e., the version of the Continuum Hypothesis for X holds. Fixing such a set X, among its uncountable subsets a notion of 'size' can be introduced from which a rich combinatorial theory can be developed. A stationary subset of X is one that is 'medium sized' with respect to this notion. A model of set theory is a collection of sets that satisfies the axioms of set theory (in the same way that 'a model of group theory' is a collection M of objects that satisfies the axioms of group theory, i.e., M is a group). Given a model of set theory V and a submodel M, some element of M may be 'stationary in M' but not be 'stationary in V'. We present two results that investigate these notions. The first says that there is always some preservation, i.e., certain stationary sets in M are stationary in V. The second studies restrictions in what M can be if an additional assumption (a so-called forcing axiom, related to preservation of stationary sets) is assumed of both M and V.
All interested persons are welcome.