Ergodic undefinability in set theory and recursion theory

Let T be a measure preserving ergodic transformation of a compact Abelian group G with normalized Haar measure m on the collection B of Borel sets; call g ∈ G generic w.r.t. a set B ∈ B iff, upon action by r, g is to stay in B with limit frequency equal to m(B). We study the definability of generic elements in Zermelo-Fraenkel set theory with Global Choice (ZFGC, which is a very good conservative extension of ZFQ, and in higher recursion theory. We prove (1) the set of those g ∈ G which are generic w.r.t. all ZFGC-definable Borel subsets of G is not ZFGC-definable, and (2) “being generic w.r.t. all hyperarithmetical properties of dyadic sequences” is not itself a hyperarithmetical property of dyadic sequences.