Bernhard Irrgang - 3 D forcing

Introduction

The basic idea is simple: I try to generalize iterated forcing. Classical iterated forcing as introduced by Solovay and Tennenbaum works with commutative systems of complete embeddings which are indexed along a well-order. In my approach, I do not consider a linear system which is indexed along a well-order but a higher-dimensional system indexed along a simplified morass.

Following this idea for finite support iterations (i.e. systems with direct limits), I started to develop a theory of so-called higher-dimensional forcing. I also found already various applications of the approach which are described the following. They are usually of the form "It is consistent that there exists a coloring such that ..." .

Motivation

Suppose we want to construct a ccc forcing ℙ of size ω₁. Then we can proceed as follows:
Let < σ_αβ : ℙ_α → ℙ_β | α < β < ω₁ > be a continuous, commutative system of complete embeddings between forcings.
Let ℙ be the direct limit of the system and assume that all ℙ_α are countable. Then every ℙ_α satifies ccc. Hence by a well-known theorem about finite support iterations ℙ also satisfies ccc. Obviously, this only works for ℙ of size ω₁ because we take a direct limit of countable structures.

What do we have to change to construct a ccc forcing ℙ of size ω₂ from countable approximations?
1. We need some structure along which we index the approximations and which replaces the ordinal ω₁ with its natural order. An appropriate structure will be given by a simplified (ω₁,1)-morass. That is, we replace the linear system by a two-dimensional one.

2. We need a replacement for the continuous, commutative system of complete embeddings. We will also construct a continuous, commutative system of embeddings. However, not all of them will be complete.

We have to assume that there exists an (ω₁,1)-morass to construct a ccc forcing ℙ of size ω₂. There exists already a very successful method to construct a ccc forcing of size ω₂ assuming only □_ω₁. This is Todorcevic’s method of ordinal walks. So, why is the new method interesting?

1. There exists a variant of it which guarantees that ℙ densely embeds into a forcing of size ω₁. Hence ℙ preserves GCH.
2. It has a natural generalization which allows to construct ccc forcings of size ω₃. Since very little is known about possible structures on ω₃, this might be interesting.

Two-dimensional applications

Three-dimensional application

Open Problems

Morasses / Example