Extender-based Magidor-Radin forcings without top extenders

TL;DR

This paper develops a new version of Extender-based Magidor-Radin forcing without top extenders, enabling novel methods to manipulate cardinal arithmetic and SCH failure on club subsets of inaccessible cardinals.

Contribution

It introduces a top-extender-free version of Extender-based Magidor-Radin forcing, offering new tools for controlling cardinal characteristics and stationary class behaviors.

Findings

Achieves SCH failure on a club subset of an inaccessible cardinal.

Constructs models with varied cardinal arithmetic on stationary classes.

Maintains cardinals and cofinalities outside the clubs.

Abstract

Continuing \cite{GitJir22}, we develop a version of Extender-based Magidor-Radin forcing where there are no extenders on the top ordinal. As an application, we provide another approach to obtain a failure of SCH on a club subset of an inaccessible cardinal, and a model where the cardinal arithmetic behaviors are different on stationary classes, whose union is the club, is provided. The cardinals and the cofinalities outside the clubs are not affected by the forcings.