In these two talks we will introduce Jensen's classes of subcomplete and subproper forcing as well as discuss some applications due to the speaker and Fuchs, and the speaker and Sakai. An important feature of proper forcing is the countable covering property: every countable set of ordinals added by a proper forcing notion is contained in a ground model countable set of ordinals. This is important in iteration theorems. Subproper forcing is a weakening of proper forcing that is still iterable while including some well known forcing notions which do add countable sets of ordinals that are not covered by anything in the ground model including Namba forcing (under CH) and Prikry forcing. One can weaken other classes of forcing notions in a similar way and the "sub"version of the countably closed forcing, known as subcomplete forcing, is a particularly interesting subclass of subproper forcing that was used by Jensen in several applications including his solution to the extended Namba problem.

In the first of these talks I will introduce the classes subproper and subcomplete forcing as well as discuss simplifications of them due to Fuchs and myself. Time permitting I will discuss new iterations theorems for these classes reminiscent of similar theorems proved for proper forcing in the context of the reals and combinatorics on \(\omega_1\) (\(\omega^\omega\)-bounding, preservation of Souslin trees etc). In the second talk I will discuss the forcing axioms for these classes including their applications and limitations. In particular, time permitting, I will discuss a recent result, joint with Hiroshi Sakai that the forcing axiom for subcomplete forcing is compatible with a \(\square_{\omega_1}\)-sequence. The take away is a class of strong forcing axioms that are compatible with a wide variety of behaviour on the level of the reals and combinatorics on cardinals below the continuum.

Students at Uni Wien are required to attend in person.