Reals n-generic relative to some perfect tree

Details Reals n-generic relative to some perfect tree Reals n-generic relative to some perfect tree

2006-10-10

arxiv.org/abs/math/0610306v2

Journal of Symbolic Logic, volume 73, June 2008, pages 401-411.

Mathematics

Logic

12 pages. Updated to final form (a few details added, minor errors corrected)

Scientific paper

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.

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