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Reals n-generic relative to some perfect tree

  • Bernard A. Anderson (a1)

Abstract

We say that a real X is n-generic relative to a perfect tree T if X is a path through T and for all sets S, there exists a number k such that either Xk ϵ S or for all a σ ϵ T extending Xk we have a σ ∉ S. A real X is n-generic relative to some perfect tree if there exists such a T. We first show that for every number n all but countably many reals are n-generic relative to some perfect tree. Second, we show that proving this statement requires ZFC + “∃ infinitely many iterates of the power set of ω”. Third, we prove that every finite iterate of the hyperjump, , is not 2-generic relative to any perfect tree and for every ordinal α below the least λ such that supβ<λ (βth admissible) = λ, the iterated hyperjump is not 5-generic relative to any perfect tree. Finally, we demonstrate some necessary conditions for reals to be 1-generic relative to some perfect tree.

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Reals n-generic relative to some perfect tree

  • Bernard A. Anderson (a1)

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