Kurt Gödel Research Center

2016 seminar talk: Forcing variations of the Martin's Axiom

Talk held by David Chodounský (Academy of Sciences of the Czech Republic, Prague) at the KGRC seminar on 2016-06-02.

Abstract

The classical technique of iterated forcing due the Solovay and Tennenbaum provides a constriction of a "$<c$ universal" c.c.c. forcing, i.e. a forcing such that in the generic extension the Martin's Axiom holds; on c.c.c. posets $<c$ generic filters exist. This motivates the following question. Given a reasonable class of (c.c.c.) forcing notions, does there exist a "$<c$ universal" forcing within this class? I will show the answer to this question is YES, the finite support iteration approach still works, but the reasoning is somewhat more involved than in the classical case.

In particular, I will prove that assuming a diamond principle on $\kappa$, given a class of c.c.c. forcings closed on finite support iterations and regular subforcings there is a forcing within this class which forces the Martin Axiom for this class together with the continuum equal to $\kappa$. The talk should be quite basic, I will review the classical method of forcing MA and I will point out the extra challenges in the general setup.