On non-wellfounded iterations of the perfect set forcing

Abstract

AbstractWe prove that ifIis a partially ordered set in a countable transitive modelof ZFC thencan be extended by a generic sequence of realsai,i∈I, such thatis preserved and everyaiis Sacks generic over[〈aj:j<i〉]. The structure of the degrees of-constructibility of reals in the extension is investigated.As applications of the methods involved, we define a cardinal invariant to distinguish product and iterated Sacks extensions, and give a short proof of a theorem (by Budinas) that in ω2-iterated Sacks extension of L the Burgess selection principle for analytic equivalence relations holds.