$^*$Forcing

Abstract

Let $M$ be a transitive model of $ZFC$ and let ${\bf B}$ be a $M$-complete Boolean algebra in $M.$ (In general a proper class.) We define a generalized notion of forcing with such Boolean algebras, $^*$forcing. (A $^*$ forcing extension of $M$ is a transitive set of the form $M[{\bf G}]$ where ${\bf G}$ is an $M$-complete ultrafilter on ${\bf B}.$) We prove that 1. If ${\bf G}$ is a $^*$forcing complete ultrafilter on ${\bf B},$ then $M[{\bf G}]\models ZFC.$ 2. Let $H\sub M.$ If there is a least transitive model $N$ such that $H\in M,$ $Ord^M=Ord^N,$ and $N\models ZFC,$ then we denote $N$ by $M[H].$ We show that all models of $ZFC$ of the form $M[H]$ are $^*$forcing extensions of $M.$ As an immediate corollary we get that $L[0^{\#}]$ is a $^*$forcing extension of $L.$