Non-Normal Magidor-Radin Types of Forcings

TL;DR

This paper develops non-normal variants of Magidor and Radin forcings, showing their properties in adding Cohen generic functions and limitations in changing cofinalities without adding subsets, with applications to Gitik's forcing.

Contribution

It introduces and analyzes non-normal Magidor and Radin forcings, extending classical results and exploring their effects on cofinalities and subsets of measurable cardinals.

Findings

Non-normal Magidor and Radin forcings can add Cohen generic functions at limit points of cofinality ω.

These forcings do not project onto κ-distributive forcings when not changing cofinality to ω₁.

A non-normal variation of Gitik's forcing can change cofinality to ω₁ while adding a Cohen subset of κ.

Abstract

We develop the non-normal variations of two classical Prikry-type forcings; namely, Magidor and Radin forcings. We generalize the fact that the non-normal Prikry forcing is a projection of the extender-based to a coordinate of the extender to our forcing and the Radin/Magidor-Radin-extender-based forcing from \cite{CarmiMagidorRadin,CarmiRadin}. Then, we show that both the non-normal variation of Magidor and Radin forcings can add a Cohen generic function to every limit point of cofinality $\omega$ of the generic club. Second, we show that this phenomenon is limited to the cases where the forcings are not designed to change the cofinality of a measurable $\kappa$ to $\omega_1$. Specifically, in the above-mentioned circumstances these forcings do not project onto any $\kappa$-distributive forcing. We use that to conclude that the extender-based Radin/Magidor-Radin forcing does not add fresh subsets to $\kappa$ as well. In the second part of the paper we focus on the natural non-normal variation of Gitik's forcing from \cite[\S3]{GitikNonStationary}. Our main result shows that this poset can be employed to change the cofinality of a measurable cardinal $\kappa$ to $\omega_1$ while introducing a Cohen subset of $\kappa$.