Logic Colloquium 2008
Katherine Thompson Obtaining inner models for global results using class forcing
The forcing technique to produce outer models has been exploited to a great extent to show an enormous range of possibilities which may be true of the universe. Relative to a fixed universe, some of the same results can be true in inner and outer models, but different techniques are used. S. Friedman introduced a programme which aims to discover what questions may be answered in an inner model. There are many consistency results achieved using forcing which are not yet known to be obtainable via inner models, since internal consistency results are harder to achieve. To obtain any consistency result, one must show that a generic filter exists. To achieve internal consistency, one needs an inner model with all the ordinals of the ambient universe, therefore one cannot restrict to a countable submodel, as is usual in set-forcing. The question then is how to build generics, if one even exists. One class of results which are studied in this context are called ``global properties''. These are statements which are true throughout the universe (e.g. at every regular cardinal). Global properties are achieved using class forcing. This adds an extra challenge to building generics as all antichains of the forcing which exist in the universe must be considered. In this talk, we will examine the techniques used to build such generics and the properties of the forcings which are sufficient to utilise these methods.