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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Ergodic undefinability in set theory and recursion theory


Author: Daniele Mundici
Journal: Proc. Amer. Math. Soc. 82 (1981), 107-111
MSC: Primary 03E47; Secondary 03E15, 22D40, 28D05
MathSciNet review: 603611
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Abstract: Let $ T$ be a measure preserving ergodic transformation of a compact Abelian group $ G$ with normalized Haar measure $ m$ on the collection $ \mathcal{B}$ of Borel sets; call $ g \in G$ generic w.r.t. a set $ B \in \mathcal{B}$ 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 \in 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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0603611-1
PII: S 0002-9939(1981)0603611-1
Article copyright: © Copyright 1981 American Mathematical Society