Reals n-generic relative to some perfect tree
Abstract
Description
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.
12 pages. Updated to final form (a few details added, minor errors corrected)
12 pages. Updated to final form (a few details added, minor errors corrected)