The Approachability Ideal Without a Maximal Set

TL;DR

This paper introduces a novel forcing technique using finite conditions and side conditions to add partial square sequences on stationary sets, demonstrating the consistency of the approachability ideal lacking a maximal set under certain large cardinal assumptions.

Contribution

It develops a new forcing method with side conditions for adding partial square sequences and shows the consistency of the approachability ideal without a maximal set assuming a Mahlo cardinal.

Findings

Forcing with finite conditions adds partial square sequences.

Certain quotients of the forcing have the $oldsymbol{ extomega_1}$-approximation property.

It is consistent that the approachability ideal $I[oldsymbol{ extomega_2}]$ lacks a maximal set assuming a Mahlo cardinal.

Abstract

We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously adding partial square sequences on multiple stationary sets. We show that certain quotients of such forcings have the $\omega_1$-approximation property. We apply these ideas to prove, assuming the consistency of a greatly Mahlo cardinal, that it is consistent that the approachability ideal $I[\omega_2]$ does not have a maximal set modulo clubs.