A MINIMAL PRIKRY-TYPE FORCING FOR SINGULARIZING A MEASURABLE CARDINAL

Abstract. Recently, Gitik, Kanovei and the first author proved that for a classical Prikry forcing extension the family of the intermediate models can be parametrized by Pp휔q{finite. By modifying the standard Prikry tree forcing we define a Prikry-type forcing which also singularizes a measurable cardinal but which is minimal, i.e. there are no intermediate models properly between the ground model and the generic extension. The proof relies on combining the rigidity of the tree structure with indiscernibility arguments resulting from the normality of the associated measures. §1. Introduction. The classical Prikry forcing first appeared in Prikry’s dis-sertation [9] in 1970. It gave a positive answer the following question of Silver and Solovay: Is there a forcing preserving all cardinals while some cofinality changes? In fact, the singularization of regular cardinals by some forcing is necessaril