Changing cofinalities and collapsing cardinals in models of set theory
The author studies the effect of extending a model \(M\) of ZFC to a model \(N\) of ZFC where the regular cardinal \(\kappa_{1}\) of \(M\) is collapsed to the cardinal \(\kappa_{0}\) in \(N\) and its new cofinality \(\rho\) is \(<\kappa_{0}\), that is \(N\vDash \text{cf}(\kappa_{1})=\rho<\kappa_{0}=|\kappa_{1}|<\kappa_{1}\). He shows that if \((^{\rho}\kappa_{0})^{N}=(^{\rho}\kappa_{0})^{M}\), then for each \(M\)-cardinal \(\lambda\) such that \(M\vDash \kappa_{1}<\lambda< \text{cc}(P(\kappa_{1})/[\kappa_{1}]^{<\kappa_{1}})\) it is the case that \(|\lambda|^{N}\leq |(\kappa_{0}^{\rho})^{M}|^{N}=(\kappa_{0}^{\rho})^{N}\). Moreover, if \((\kappa_{0}^{\rho})^{N}=\kappa_{0}\) then \(|\lambda|^{N}=\kappa_{0}\). For \(N=M[f]\), where \(f\colon\rho\rightarrow\kappa_{1}\) is an unbounded mapping, \(N\) is a \(|\lambda|=\kappa_{0}\)-minimal extension. Applications are made to the forcing notions of Bukovský and Namba.
scientific article; zbMATH DE number 1882257