Kurt Gödel Research Center

2015 seminar talk: Non-pretame Class Forcing and the Forcing Theorem, the axioms of ZFC and non-definable Class Forcing

Talk held by Peter Holy (Universität Bonn, Germany) at the KGRC seminar on 2015-05-27.

Abstract

Pretameness is a combinatorial condition on class forcings that was introduced by Sy Friedman and that is equivalent to the preservation of the axioms of ZF^-, that is ZF without the power set axiom. Moreover pretameness implies the forcing theorem to hold, that is, the definability of the forcing relation and the truth lemma (the latter states that everything true in a generic extension is forced by some condition in the generic).

Building on an unpublished result of Sy Friedman, we show that there is always a (non-pretame) class forcing that fails to satisfy the forcing theorem, and that consistently, there is one that even fails to satisfy the truth lemma.

We observe that all “simple” examples of non-pretame class forcings do not only destroy instances of the axiom of replacement, but also of the axiom of separation. We conjecture that this is actually the case for all (definable) class forcings, however we give an example of a non-definable class forcing, a class sized version of Prikry forcing, that destroys replacement, however preserves all instances of separation.

This is joint work with Regula Krapf, Philipp Lücke, Ana Njegomir and Philipp Schlicht.