Perfect-set forcing for uncountable cardinals

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