Reals n-generic relative to some perfect tree

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 Σ⁰_n (T) sets S, there exists a number k such that either X|k ∈ S or for all σ ∈ T extending X|k we have 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, 𝒪⁽ⁿ⁾, is not 2-generic relative to any perfect tree and for every ordinal α below the least λ such that sup_β<λ (βth admissible) = λ, the iterated hyperjump 𝒪^(α) 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.