Forcing with overlapping supercompact extenders

TL;DR

This paper develops a supercompact forcing method that can enlarge the power set of certain singular cardinals while preserving their cardinality and collapsing their successors, extending previous constructions.

Contribution

It introduces a new supercompact forcing technique that allows for controlling the size of power sets of singular cardinals with specific preservation properties.

Findings

Enlarges the power set of singular cardinals arbitrarily.

Preserves the cardinality of the singular cardinal.

Collapses the successor of the singular cardinal while preserving higher cardinals.

Abstract

We build a supercompact version of the forcing defined in \cite{gitik2019}. For each singular cardinal in the ground model with any fixed cofinality, which is a limit of supercompact cardinals, it is possible to force so that the size of the powerset of the singular cardinal is arbitrarily large, while preserving the singular cardinal. An important feature of this forcing is that it is possible to define the forcing so that the successor of the singular cardinal is collapsed, but all the cardinals above it are preserved.