Disjoint Stationary Sequences on an Interval of Cardinals

TL;DR

This paper constructs a model with disjoint stationary sequences on certain cardinals and distinguishes between internally stationary and internally club sets, answering questions in set theory using advanced forcing techniques.

Contribution

It introduces a new model demonstrating disjoint stationary sequences on $eth_{n+2}$ and differentiates internal stationarity and clubness on a stationary set, solving open questions.

Findings

Existence of disjoint stationary sequences on $eth_{n+2}$ for all $n$

Distinction between internally stationary and internally club sets

Use of Mitchell forcing variants with specific support methods

Abstract

We answer a question of Krueger by obtaining -- from countably many Mahlo cardinals -- a model where there is a disjoint stationary sequence on $\aleph_{n+2}$ for every $n\in\omega$. In that same model, the notions of being internally stationary and internally club are distinct on a stationary subset of $[H(\Theta)]^{\aleph_{n+1}}$ for every $n\in\omega$ and $\Theta\geq\aleph_{n+2}$, answering another of Krueger's questions. This is obtained by employing a product of variants of Mitchell forcing which uses finite support for the Cohen reals and full support for the countably many collapses.