Successors of singular cardinals and measurability
In the presence of the axiom of choice measurable cardinals must be very big, but where choice fails, such as in the world of AD, small cardinals such as \(\aleph_ 1\), \(\aleph_ 2\) can be measurable. Earlier relative consistency results of Jech and Takeuti produced inner models where the successor of a regular cardinal can be made measurable. Apter wants to do the same for successors of singular cardinals. Under certain strong hypotheses he succeeds. Theorem 1. Assume \(V\vDash ''ZFC\) \(+\kappa <\lambda\) are cardinals where \(\lambda\) is measurable and \(\kappa\) is \(2^{\lambda}\) supercompact''. Then there is a model of ''ZF \(+\) \(AC_{\kappa}\) \(+\kappa\) is a strong limit cardinal \(+\) \(cof(\kappa)=\omega\) \(+\kappa\) is a Rowbottom cardinal \(+\kappa^+\) is measurable''. The proof uses a generalization of Prikry forcing. Then by a collapsing argument Apter proves (under the same hypotheses as Theorem 1) that there is a model of ''ZF \(+\aleph_{\omega}\) is a Rowbottom cardinal \(+\aleph_{\omega +1}\) is a measurable cardinal''.
scientific article; zbMATH DE number 3995664