Higher-dimensional forcing

TL;DR

This paper discusses a method for constructing larger ccc forcings using higher-dimensional systems indexed along simplified morasses, extending the classical approach based on linear systems and finite support iterations.

Contribution

It introduces a novel approach to build larger ccc forcings by employing higher-dimensional indexing with simplified morasses, expanding the scope of forcing constructions.

Findings

Larger ccc forcings can be constructed using higher-dimensional systems.

The method extends classical finite support iteration techniques.

This approach enables new possibilities in forcing theory.

Abstract

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 $\omega_1$ is given. Then its direct limit satisfies ccc by a well-known theorem on finite support iterations. However, this limit has size at most $\omega_1$. To get larger forcings, we do not consider linear systems but higher-dimensional systems which are indexed along simplified morasses.