Coherent systems of finite support iterations

TL;DR

This paper introduces 3D-coherent systems of finite support iterations to construct models with specific configurations in Cichoń's diagram, including a separation into seven distinct values and a compatible well-order of the reals.

Contribution

It develops a new forcing technique using 3D-coherent systems of FS iterations to produce complex models in set theory.

Findings

Constructed a model with 7 distinct values in Cichoń's diagram.

Established the consistency of a 7-value constellation with a $^1_3$ well-order of the reals.

Demonstrated the effectiveness of 3D-coherent systems in set-theoretic constructions.

Abstract

We introduce a forcing technique to construct three-dimensional arrays of generic extensions through FS (finite support) iterations of ccc posets, which we refer to as 3D-coherent systems. We use them to produce models of new constellations in Cicho\'n's diagram, in particular, a model where the diagram can be separated into 7 different values. Furthermore, we show that this constellation of 7 values is consistent with the existence of a $\Delta^1_3$ well-order of the reals.