211 research outputs found

    Incompleteness and jump hierarchies

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    This paper is an investigation of the relationship between G\"odel's second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector's that the relation {(A,B)∈R2:OA≤HB}\{(A,B) \in \mathbb{R}^2 : \mathcal{O}^A \leq_H B\} is well-founded. We provide an alternative proof of this fact that uses G\"odel's second incompleteness theorem instead of the theory of admissible ordinals. We then derive a semantic version of the second incompleteness theorem, originally due to Mummert and Simpson, from this result. Finally, we turn to the calculation of the ranks of reals in this well-founded relation. We prove that, for any A∈RA\in\mathbb{R}, if the rank of AA is α\alpha, then ω1A\omega_1^A is the (1+α)th(1 + \alpha)^{\text{th}} admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of XX is ω1X\omega_1^X.Comment: 11 pages. Corrects a mistake in the statements of two result

    Tree-Automatic Well-Founded Trees

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    We investigate tree-automatic well-founded trees. Using Delhomme's decomposition technique for tree-automatic structures, we show that the (ordinal) rank of a tree-automatic well-founded tree is strictly below omega^omega. Moreover, we make a step towards proving that the ranks of tree-automatic well-founded partial orders are bounded by omega^omega^omega: we prove this bound for what we call upwards linear partial orders. As an application of our result, we show that the isomorphism problem for tree-automatic well-founded trees is complete for level Delta^0_{omega^omega} of the hyperarithmetical hierarchy with respect to Turing-reductions.Comment: Will appear in Logical Methods of Computer Scienc

    Finding subsets of positive measure

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    An important theorem of geometric measure theory (first proved by Besicovitch and Davies for Euclidean space) says that every analytic set of non-zero ss-dimensional Hausdorff measure Hs\mathcal H^s contains a closed subset of non-zero (and indeed finite) Hs\mathcal H^s-measure. We investigate the question how hard it is to find such a set, in terms of the index set complexity, and in terms of the complexity of the parameter needed to define such a closed set. Among other results, we show that given a (lightface) Σ11\Sigma^1_1 set of reals in Cantor space, there is always a Π10(O)\Pi^0_1(\mathcal{O}) subset on non-zero Hs\mathcal H^s-measure definable from Kleene's O\mathcal O. On the other hand, there are Π20\Pi^0_2 sets of reals where no hyperarithmetic real can define a closed subset of non-zero measure.Comment: This is an extended journal version of the conference paper "The Strength of the Besicovitch--Davies Theorem". The final publication of that paper is available at Springer via http://dx.doi.org/10.1007/978-3-642-13962-8_2

    The weakness of the pigeonhole principle under hyperarithmetical reductions

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    The infinite pigeonhole principle for 2-partitions (RT21\mathsf{RT}^1_2) asserts the existence, for every set AA, of an infinite subset of AA or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that RT21\mathsf{RT}^1_2 admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every Δn0\Delta^0_n set, of an infinite lown{}_n subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak, Jockusch and Slaman.Comment: 29 page

    Banach Spaces as Data Types

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    We introduce the operators "modified limit" and "accumulation" on a Banach space, and we use this to define what we mean by being internally computable over the space. We prove that any externally computable function from a computable metric space to a computable Banach space is internally computable. We motivate the need for internal concepts of computability by observing that the complexity of the set of finite sets of closed balls with a nonempty intersection is not uniformly hyperarithmetical, and thus that approximating an externally computable function is highly complex.Comment: 20 page
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