Some basic thoughts on the cofinalities of Chang structures with an application to forcing

Abstract

Consider false(κ+++,κ++false)↠false(κ+,κfalse) where κ is an uncountable regular cardinal. By a result of Shelah's we have cof(X∩κ++)=κ for almost all X⊂κ+++ witnessing this. Here we consider the question if there could be a similar result for X∩κ+. We will discuss some basic facts implying that this cannot hold in general. We will use these facts to construct an interesting example of a pseudo Prikry forcing, answering a question of Sinapova.