Higher randomness and forcing with closed sets

Author Benoit Monin

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Benoit Monin

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Benoit Monin. Higher randomness and forcing with closed sets. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 566-577, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


[Kechris, Trans. Amer. Math. Soc. 1975] showed 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. London Math. Soc. 2007]. Due to its universal nature, it was conjectured by many that this nullset has a high Borel rank (the question is explicitely mentioned by Chong and Yu, and in [Yu, Fund. Math. 2011]). In this paper, we refute this conjecture and show that this nullset is merely Sigma_3^0. Together with a result of Liang Yu, our result also implies that the exact Borel complexity of this set is Sigma_3^0. 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.
  • Effective descriptive set theory
  • Higher computability
  • Effective randomness
  • Genericity


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