Logic & Set Theory seminar: The Dark Side of Class Forcing
Peter Holy, University of Bonn
Howard House, 4th floor seminar room
Class Forcing is the generalization of set forcing where we relax the condition that forcing posets have to be sets, but allow for them to be proper classes, and require generic filters to intersect all dense subclasses of the forcing poset (rather than all dense subsets of the forcing poset, as we do in set forcing). While it is known that class forcing (unlike set forcing) need not preserve the axioms of ZFC (in particular that it can destroy Replacement and Separation), we present several results that show how badly class forcing can in fact behave. Our main results concern failures of the forcing theorem, the central theorem about (set) forcing, stating that the forcing relation is definable and that the truth lemma holds, that is: anything true in a generic extension is forced by some condition in the forcing. The forcing theorem holds for all ZFC-preserving class forcings, and it also holds for many class forcings that destroy ZFC. We will present quite a weak condition on class forcings that implies the forcing theorem to hold, and also present the following results (which, to be precise, refer to countable transitive models of ZFC):
* ) There is a class forcing, a first order formula phi and names sigma and tau such that the collection of conditions p forcing that sigma=tau is not definable.
* ) It is consistent that there is a class forcing for which the above collection of conditions is not amenable.
* ) If Morse-Kelley class theory is consistent, then so is ZFC together with the existence of a class forcing for which the Truth Lemma fails.
This is joint work with Regula Krapf, Philipp Luecke, Ana Njegomir and Philipp Schlicht.
Organiser: Andrew Brooke-Taylor