We generalize the notion of finite support iteration of forcings. That is, instead of considering linear systems of forcings and taking direct limits, we consider higher-dimensional systems and take the morass limit. Like in the case of normal finite support iterations, chain conditions are preserved.
One application is a ccc-forcing of size ω_{1} that adds an ω_{2}-Suslin tree. Another is a ccc-forcing of size ω_{1} that adds a 0-dimensional Hausdorff topology on ω_{3} which has spread ω_{1}. An example for a forcing that necessarily destroys GCH is the ccc forcing which adds a strong chain in P(ω_{1}) mod finite.
We give the definition of a simplified gap-1 morass and prove some of its basic properties.
We introduce FS iterations along gap-1 morasses and show that they preserve ccc.
We construct along a gap-1 morass a ccc forcing that adds an ω_{2}-Suslin tree.
We introduce the notion of a simplified gap-2 morass and summarize its basic properties.
We continue our summary of gap-2 morass properties and define the notion of FS iteration along a gap-2 morass.
This session had to be cancelled. There was an Oberseminar talk by Russel Miller instead.
We apply our methods to Cohen-forcing. This yields a ccc-forcing of size ω_{1} that adds a 0-dimensional Hausdorff topology on ω_{3} which has spread ω_{1}.