Diagonal supercompact Radin forcing

TL;DR

This paper introduces a novel forcing method called diagonal supercompact Radin forcing to construct models with many singular cardinals where certain combinatorial principles fail, addressing open questions in set theory.

Contribution

It develops a new forcing technique that enables the creation of models with specific properties regarding singular cardinals and combinatorial principles.

Findings

Constructed a model with many singular cardinals where SCH fails

Produced a model where weak square principle fails at many cardinals

Demonstrated the consistency of these properties with large cardinal assumptions

Abstract

Motivated by the goal of constructing a model in which there are no $\kappa$-Aronszajn trees for any regular $\kappa>\aleph_1$, we produce a model with many singular cardinals where both the singular cardinals hypothesis and weak square fail.