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

Abstract

Consider $({\kappa }^{+++},{\kappa }^{++})\twoheadrightarrow ({\kappa }^{+},\kappa )$ where $\kappa$ is an uncountable regular cardinal. By a result of Shelah's we have $\mathrm{cof}(X\cap {\kappa }^{++})=\kappa$ for almost all $X\subset {\kappa }^{+++}$ witnessing this. Here we consider the question if there could be a similar result for $X\cap {\kappa }^{+}$. We use this discussion to give an interesting example of a pseudo Prikry forcing answering a question of Sinapova.