Singularizing successor cardinals by forcing

Abstract

We exhibit models of set theory, using large cardinals and forcing, in which successor cardinals can be made singular by some “Namba-like” further set forcing, in most cases without collapsing cardinals below that successor cardinal. For successors of regular cardinals, we work from consistency-wise optimal assumptions in the ground model. Successors of singular cardinals require stronger hypotheses. Our partial orderings are different from Woodin’s stationary tower forcing, which requires much stronger hypotheses when singularizing successors of regular cardinals, and collapses cardinals above the cardinal whose cofinality is changed when singularizing successors of singular cardinals.