On disjoint stationary sequences

TL;DR

This paper constructs disjoint stationary sequences on successive cardinals and explores conditions under which such sequences exist or do not exist, using advanced forcing techniques and large cardinal assumptions.

Contribution

It provides an alternative presentation of mixed support iteration as Mitchell forcing and answers open questions about the existence of disjoint stationary sequences.

Findings

Constructed disjoint stationary sequences on successive cardinals.

Showed a model where  is not in I[] and no disjoint stationary sequence exists on .

Used Mahlo cardinals to establish consistency results.

Abstract

We answer a question of Krueger by obtaining disjoint stationary sequences on successive cardinals. The main idea is an alternative presentation of a mixed support iteration, using it even more explicitly as a variant of Mitchell forcing. We also use a Mahlo cardinal to obtain a model in which $\aleph_2 \notin I[\aleph_2]$ and there is no disjoint stationary sequence on $\aleph_2$, answering a question of Gilton.