# 2010 seminar talk: Small proper forcing and the size of the continuum

Talk held by Miguel Angel Mota (KGRC) at the KGRC seminar on 2010-12-02.

### Abstract

In a recent work with David Asperó I proved that the forcing axiom for the class of all the proper posets of small cardinality does not impose any bound on the size of he continuum. The corresponding proof is quite technical and uses some new ideas regarding forcing iteration. During this talk, however, I will prove something more modest: the consistency of the forcing axiom for the class of all the finitely proper posets together with a large continuum. On the one hand, this will give me the opportunity to explain the main ingredients of our main result. On the other, this is near to be optimal since it is consistent to assume that all the small proper posets are finitely proper.