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ABsTRAcr. Let T be a measure preserving ergodic transformation of a compact Abelian group G with normalized Haar measure m on the collection ' of Borel sets; call g E G generic w.r.t. a set B E ' iff, upon action by T, 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 ZFC), and in higher recursion theory. We prove (1) the set of those g E 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.

Ergodic undefinability in set theory and recursion theory