183 research outputs found

    Grilliot's trick in Nonstandard Analysis

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    The technique known as Grilliot's trick constitutes a template for explicitly defining the Turing jump functional (∃2)(\exists^2) in terms of a given effectively discontinuous type two functional. In this paper, we discuss the standard extensionality trick: a technique similar to Grilliot's trick in Nonstandard Analysis. This nonstandard trick proceeds by deriving from the existence of certain nonstandard discontinuous functionals, the Transfer principle from Nonstandard analysis limited to Π10\Pi_1^0-formulas; from this (generally ineffective) implication, we obtain an effective implication expressing the Turing jump functional in terms of a discontinuous functional (and no longer involving Nonstandard Analysis). The advantage of our nonstandard approach is that one obtains effective content without paying attention to effective content. We also discuss a new class of functionals which all seem to fall outside the established categories. These functionals directly derive from the Standard Part axiom of Nonstandard Analysis.Comment: 21 page

    Reverse Mathematics and parameter-free Transfer

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    Recently, conservative extensions of Peano and Heyting arithmetic in the spirit of Nelson's axiomatic approach to Nonstandard Analysis, have been proposed. In this paper, we study the Transfer axiom of Nonstandard Analysis restricted to formulas without parameters. Based on this axiom, we formulate a base theory for the Reverse Mathematics of Nonstandard Analysis and prove some natural reversals, and show that most of these equivalences do not hold in the absence of parameter-free Transfer.Comment: 22 pages; to appear in Annals of Pure and Applied Logi

    The computational content of Nonstandard Analysis

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    Kohlenbach's proof mining program deals with the extraction of effective information from typically ineffective proofs. Proof mining has its roots in Kreisel's pioneering work on the so-called unwinding of proofs. The proof mining of classical mathematics is rather restricted in scope due to the existence of sentences without computational content which are provable from the law of excluded middle and which involve only two quantifier alternations. By contrast, we show that the proof mining of classical Nonstandard Analysis has a very large scope. In particular, we will observe that this scope includes any theorem of pure Nonstandard Analysis, where `pure' means that only nonstandard definitions (and not the epsilon-delta kind) are used. In this note, we survey results in analysis, computability theory, and Reverse Mathematics.Comment: In Proceedings CL&C 2016, arXiv:1606.0582

    Pincherle's theorem in Reverse Mathematics and computability theory

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    We study the logical and computational properties of basic theorems of uncountable mathematics, in particular Pincherle's theorem, published in 1882. This theorem states that a locally bounded function is bounded on certain domains, i.e. one of the first 'local-to-global' principles. It is well-known that such principles in analysis are intimately connected to (open-cover) compactness, but we nonetheless exhibit fundamental differences between compactness and Pincherle's theorem. For instance, the main question of Reverse Mathematics, namely which set existence axioms are necessary to prove Pincherle's theorem, does not have an unique or unambiguous answer, in contrast to compactness. We establish similar differences for the computational properties of compactness and Pincherle's theorem. We establish the same differences for other local-to-global principles, even going back to Weierstrass. We also greatly sharpen the known computational power of compactness, for the most shared with Pincherle's theorem however. Finally, countable choice plays an important role in the previous, we therefore study this axiom together with the intimately related Lindel\"of lemma.Comment: 43 pages, one appendix, to appear in Annals of Pure and Applied Logi

    The proof-theoretic strength of Ramsey's theorem for pairs and two colors

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    Ramsey's theorem for nn-tuples and kk-colors (RTkn\mathsf{RT}^n_k) asserts that every k-coloring of [N]n[\mathbb{N}]^n admits an infinite monochromatic subset. We study the proof-theoretic strength of Ramsey's theorem for pairs and two colors, namely, the set of its Π10\Pi^0_1 consequences, and show that RT22\mathsf{RT}^2_2 is Π30\Pi^0_3 conservative over IΣ10\mathsf{I}\Sigma^0_1. This strengthens the proof of Chong, Slaman and Yang that RT22\mathsf{RT}^2_2 does not imply IΣ20\mathsf{I}\Sigma^0_2, and shows that RT22\mathsf{RT}^2_2 is finitistically reducible, in the sense of Simpson's partial realization of Hilbert's Program. Moreover, we develop general tools to simplify the proofs of Π30\Pi^0_3-conservation theorems.Comment: 32 page
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