12,567 research outputs found

    Higher randomness and forcing with closed sets

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    [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 Genericity and Differentiability

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    We prove that a real x is 1-generic if and only if every differentiable computable function has continuous derivative at x. This provides a counterpart to recent results connecting effective notions of randomness with differentiability. We also consider multiply differentiable computable functions and polynomial time computable functions.Comment: Revision: added sections 6-8; minor correction

    Open questions about Ramsey-type statements in reverse mathematics

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    Ramsey's theorem states that for any coloring of the n-element subsets of N with finitely many colors, there is an infinite set H such that all n-element subsets of H have the same color. The strength of consequences of Ramsey's theorem has been extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility. Our understanding of the combinatorics of Ramsey's theorem and its consequences has been greatly improved over the past decades. In this paper, we state some questions which naturally arose during this study. The inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey's theorem.Comment: 15 page
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