Intermediate Models of Magidor-Radin Forcing- Part II

TL;DR

This paper extends the understanding of Magidor-Radin forcing by showing that certain sets in generic extensions are definable from specific subsets and analyzing how various forcing notions affect cofinality.

Contribution

It proves that in Magidor-Radin generic extensions, sets of ordinals are definable from subsets of the generic club, and it characterizes how cofinalities change under different forcing notions.

Findings

Sets of ordinals in the extension are definable from subsets of the generic club.

The supremum of a fresh set in various forcings changes cofinality to .

The results generalize previous work on Magidor-Radin forcing.

Abstract

We continue the work done by the authors and before that by the second author, Kanovei and koepke. We prove that for every set of ordinals $A$ in a Magidor-Radin generic extension using a coherent sequence such that $o^{\vec{U}}(\kappa)<\kappa^+$, there is $C'\subseteq C_G$, such that $V[A]=V[C']$. Also we prove that the supremum of a fresh set in a Prikry, tree Prikry, Magidor, Radin-Magidor and Radin forcing, changes cofinality to $\omega$.