Sigma-Prikry forcing III: Down to Aleph_omega

TL;DR

This paper proves the consistency of the failure of the singular cardinals hypothesis at leph_omega and the reflection of all stationary subsets of leph_{omega+1}, showing two classic results can coexist.

Contribution

It demonstrates the simultaneous consistency of two longstanding results in set theory using advanced forcing techniques.

Findings

Failure of the SCH at leph_omega is consistent.

All stationary subsets of leph_{omega+1} reflect simultaneously.

Magidor's results from 1977 and 1982 can both hold together.

Abstract

We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_\omega$ together with the reflection of all stationary subsets of $\aleph_{\omega+1}$. This shows that two classic results of Magidor (from 1977 and 1982) can hold simultaneously.