1992/09/25 by Garvin Melles, Melles, Garvin
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.math/9209210
openalex publication_date 1992/09/25 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28
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∈ M, OrdM=OrdN, 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.