Perfect-set forcing for uncountable cardinals

Abstract

Perfect-set forcing has been around for a long time. Sacks [10] himself had made substantial use of it to get important minimality results both in set theory and in recursion theory, and the fusion idea that he popularized has become an integral part of several notions of forcing. After Laver [8] developed the idea of adding reals iteratively with countable support, Baumgartner and Laver [2] applied it to the case of perfect-set forcing to produce interesting consistency results about Ramsey ultrafilters over to and the tree property for co 2. Since then, work of Shelah, Baumgartner, and others has considerably systematized countable support iterated forcing. As a first step in generalization, I develop in this paper a notion of perfect-set forcing for regular uncountable cardinals K and its iteration with K size supports. An application of an effective version of this forcing has already been made in recent work by Sacks and Slaman [11] in the study of abstract E-recursion and sideways extensions of E-closed structures. in Section I the notion of forcing and its iteration are formulated, and their basic properties established. In particular, the appropriate fusion lemmas are stated and proved. Section 2 is dominated by the long proof of a key technical theorem, one of whose many consequences is that ~ + is preserved as a cardinal by the iterated forcing. The use of a ~ sequence in the ground model is an essential feature of this fusion argument. There is much less control over the forcing machinery in the uncountable case as compared to the to case considered in [2], but K gives us just enough structural information about subsets of K to allow more economical procedures to work. In fact, it will be clear that this paper owes an obvious debt to [2]. with the new modulations arising primarily from limit stage constructions and the use of O~. in Section 3 it is shown that if 2 ~ = ~*, then ~ K + had been satisfied in the ground model. In Section 4, the result on Aronszajn trees in [2] is lifted: Using K, then there are no K++-Aronszajn trees in the resulting extension.