Mitchell's Theorem Revisited

TL;DR

This paper provides a new proof of Mitchell's theorem on the approachability ideal using an abstract side condition framework, demonstrating its consistency relative to a greatly Mahlo cardinal.

Contribution

It introduces an abstract side condition method to reprove Mitchell's theorem, expanding the toolkit for set-theoretic consistency results.

Findings

New proof of Mitchell's theorem established

Approachability ideal's properties clarified

Framework applicable to other set-theoretic consistency proofs

Abstract

Mitchell's theorem on the approachability ideal states that it is consistent relative to a greatly Mahlo cardinal that there is no stationary subset of $\omega_2 \cap \mathrm{cof}(\omega_1)$ in the approachability ideal $I[\omega_2]$. In this paper we give a new proof of Mitchell's theorem, deriving it from an abstract framework of side condition methods.