Theory of Computing Systems

, Volume 60, Issue 3, pp 421–437 | Cite as

Higher Randomness and Forcing with Closed Sets

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Abstract

Kechris showed in Kechris (Trans. Am. Math. Soc. 202, 259–297, 1975) that there exists a largest \({\Pi ^{1}_{1}}\) set of measure 0. An explicit construction of this largest \({\Pi ^{1}_{1}}\) nullset has later been given in Hjorth and Nies (J. Lond. Math. Soc. 75(2), 495–508, 2007). Due to its universal nature, it was conjectured by many that this nullset has a high Borel rank (the question is explicitely mentioned in Chong and Yu (J. Symb. Log. 80(04), 1131–1148, 2015) and Yu (Fundam. Math. 215, 219–231, 2011)). In this paper, we refute this conjecture and show that this nullset is merely \({\Sigma }^{0}_{3}\). Together with a result of Liang Yu, our result also implies that the exact Borel complexity of this set is \({\Sigma }^{0}_{3}\). To do this proof, we develop the machinery of effective randomness and effective Solovay genericity, investigating the connections between those notions and effective domination properties.

Keywords

Effective descriptive set theory Higher computability Effective randomness Genericity 

Notes

Acknowledgments

I would like to thank Laurent Bienvenu, Noam Greenberg, Paul Shafer and Liang Yu for helpful comments and discussions.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.LACL LaboratoryUniversity of Paris-EstCréteilFrance

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