Reals n-generic relative to some perfect tree

Bernard A. Anderson

Published 2006-10-10, updated 2007-11-01Version 2

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 Sigma^0_n (T) sets S, there exists a number k such that either X|k is in S or for all tau in T extending X|k we have tau is not in 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^- + ``There exist infinitely many iterates of the power set of the natural numbers''. 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 alpha below the least lambda such that sup_{beta < lambda} (beta th admissible) = lambda, the iterated hyperjump O^(alpha) 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.