Applications of iterated perfect set forcing
Perfect set forcing has been used by Sacks to generate a model of ZF with a minimal L-degree. The author improves on the method of iterated perfect set forcing to encode information on the generic sequence into the partial order on the L-degrees (or, more generally, M-degrees for M an inner model of ZF), the problem being that the degree ordering on the reals is absolute while the generic sequence choosing one real from each degree is not. Her strategy is designed to obtain results both on the partial ordering of the M-degrees and the partial ordering \({\mathcal M}[\Phi]\) of countable transitive models of \(ZF+\Phi\) of the same height. Applications yield a minimal model for \(\neg CH\) (while Sacks' method produces a minimal model of \(V\neq L)\) and a model with a real minimal w.r.t. collapsing cardinals. Concerning the partial ordering \({\mathcal M}[\Phi]\) it is shown that \({\mathcal M}[AC]\) admits no non-trivial automorphism, \({\mathcal M}[\neg CH]\) and \({\mathcal M}[V\neq L]\) are not isomorphic. Added are two appendices on the existence of minimal models for Martin's Axiom MA (a result due to Shelah) and for \(\neg CH+MA\) resp. (a theorem of Woodin).
scientific article; zbMATH DE number 4061227
scientific article