arXiv subjects
Publications and source records attributed to Bernhard Irrgang.
We prove that if there exists a simplified $(ω_1,2)$-morass, then there is a ccc forcing which adds an $ω_3$-chain in P($ω_1$) mod finite and a ccc forcing which adds a family of $ω_3$-many strongly almost disjoint functions from $ω_1$ to $ω$. The idea is to use a finite support iteration of countable forcings which is not linear but three-dimensional.
In a previous paper, we introduced a way of constructing a forcing along a simplified gap-1 morass such that the forcing satisfies a chain condition. Now, we generalize this to gap-2 morasses. As an application, we prove that GCH is consistent with the existence of a 0-dimensional Hausdorff topology on $ω_3$ which has spread $ω_1$.
In a previous paper I proposed a notion of $(ω_1,β)$-morasses for $ω_1 \leq β$. In the present paper such morasses are constructed in an inner model which satisfies amenability, coherence and condensation.
I propose a notion of $(ω_1,β)$-morass for the case $ω_1 \leq β$.
This is an overview about a method of constructing ccc forcings: Suppose first that a continuous, commutative system of complete embeddings between countable forcings indexed along $ω_1$ is given. Then its direct limit satisfies ccc by a well-known theorem on finite support iterations. However, this limit has size at most $ω_1$. To get larger forcings, we do not consider linear systems but higher-dimensional systems which are indexed along simplified morasses.
We introduce a method of constructing a forcing along a simplified $(κ,1)$-morass such that the forcing satisfies the $κ$-chain condition. Alternatively, this may be seen as a method to thin out a larger forcing to get a chain condition. As an application, we construct a ccc forcing that adds an $ω_2$-Suslin tree. Related methods are Shelah's historic forcing and Todorcevic's $ρ$-functions.