54 found
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  1.  36
    On the mathematical and foundational significance of the uncountable.Dag Normann & Sam Sanders - 2019 - Journal of Mathematical Logic 19 (1):1950001.
    We study the logical and computational properties of basic theorems of uncountable mathematics, including the Cousin and Lindelöf lemma published in 1895 and 1903. Historically, these lemmas were among the first formulations of open-cover compactness and the Lindelöf property, respectively. These notions are of great conceptual importance: the former is commonly viewed as a way of treating uncountable sets like e.g. [Formula: see text] as “almost finite”, while the latter allows one to treat uncountable sets like e.g. [Formula: see text] (...)
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  2.  13
    Computability theory, nonstandard analysis, and their connections.Dag Normann & Sam Sanders - 2019 - Journal of Symbolic Logic 84 (4):1422-1465.
    We investigate the connections between computability theory and Nonstandard Analysis. In particular, we investigate the two following topics and show that they are intimately related. A basic property of Cantor space$2^ $ is Heine–Borel compactness: for any open covering of $2^ $, there is a finite subcovering. A natural question is: How hard is it to compute such a finite subcovering? We make this precise by analysing the complexity of so-called fan functionals that given any $G:2^ \to $, output a (...)
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  3.  17
    Pincherle's theorem in reverse mathematics and computability theory.Dag Normann & Sam Sanders - 2020 - Annals of Pure and Applied Logic 171 (5):102788.
    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 (...)
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  4.  17
    The strength of compactness in Computability Theory and Nonstandard Analysis.Dag Normann & Sam Sanders - 2019 - Annals of Pure and Applied Logic 170 (11):102710.
  5.  10
    On the Uncountability Of.Dag Normann & Sam Sanders - 2022 - Journal of Symbolic Logic 87 (4):1474-1521.
    Cantor’s first set theory paper (1874) establishes the uncountability of ${\mathbb R}$. We study this most basic mathematical fact formulated in the language of higher-order arithmetic. In particular, we investigate the logical and computational properties of ${\mathsf {NIN}}$ (resp. ${\mathsf {NBI}}$ ), i.e., the third-order statement there is no injection resp. bijection from $[0,1]$ to ${\mathbb N}$. Working in Kohlenbach’s higher-order Reverse Mathematics, we show that ${\mathsf {NIN}}$ and ${\mathsf {NBI}}$ are hard to prove in terms of (conventional) comprehension axioms, (...)
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  6.  29
    Set recursion and Πhalf-logic.Jean-Yves Girard & Dag Normann - 1985 - Annals of Pure and Applied Logic 28 (3):255-286.
  7.  18
    Algebraic Recursion Theory.Dag Normann - 1988 - Journal of Symbolic Logic 53 (3):986-987.
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  8.  32
    Total objects in inductively defined types.Lill Kristiansen & Dag Normann - 1997 - Archive for Mathematical Logic 36 (6):405-436.
    Coherence-spaces and domains with totality are used to give interpretations of inductively defined types. A category of coherence spaces with totality is defined and the closure of positive inductive type constructors is analysed within this category. Type streams are introduced as a generalisation of types defined by strictly positive inductive definition. A semantical analysis of type streams with continuous recursion theorems is established. A hierarchy of domains with totality defined by positive induction is defined, and density for a sub-hierarchy is (...)
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  9.  16
    The continuous functionals; computations, recursions and degrees.Dag Normann - 1981 - Annals of Mathematical Logic 21 (1):1.
  10.  9
    Recursion on the countable functionals.Dag Normann - 1980 - New York: Springer Verlag.
  11.  14
    The 1-section of a countable functional.Dag Normann & Stan S. Wainer - 1980 - Journal of Symbolic Logic 45 (3):549-562.
  12.  7
    The Biggest Five of Reverse Mathematics.Dag Normann & Sam Sanders - forthcoming - Journal of Mathematical Logic.
    The aim of Reverse Mathematics (RM for short) is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent to the theorem, working over the base theory of RM, a weak system of computable mathematics. The Big Five phenomenon of RM is the observation that a large number of theorems from ordinary mathematics are either provable in the base theory or equivalent to one of only four systems; these five (...)
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  13.  66
    Countable functionals and the projective hierarchy.Dag Normann - 1981 - Journal of Symbolic Logic 46 (2):209-215.
  14. Computing with functionals: Computability theory or computer science?Dag Normann - 2006 - Bulletin of Symbolic Logic 12 (1):43-59.
    We review some of the history of the computability theory of functionals of higher types, and we will demonstrate how contributions from logic and theoretical computer science have shaped this still active subject.
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  15.  67
    Computability over the Partial Continuous Functionals.Dag Normann - 2000 - Journal of Symbolic Logic 65 (3):1133-1142.
    We show that to every recursive total continuous functional $\Phi$ there is a PCF-definable representative $\Psi$ of $\Phi$ in the hierarchy of partial continuous functionals, where PCF is Plotkin's programming language for computable functionals. PCF-definable is equivalent to Kleene's S1-S9-computable over the partial continuous functionals.
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  16.  22
    Computability in Europe 2011.Sam Buss, Benedikt Löwe, Dag Normann & Ivan Soskov - 2013 - Annals of Pure and Applied Logic 164 (5):509-510.
  17.  26
    Embeddability of ptykes.Jean-Yves Girard & Dag Normann - 1992 - Journal of Symbolic Logic 57 (2):659-676.
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  18.  17
    Models for recursion theory.Johan Moldestad & Dag Normann - 1976 - Journal of Symbolic Logic 41 (4):719-729.
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  19.  12
    A continuous functional with noncollapsing hierarchy.Dag Normann - 1978 - Journal of Symbolic Logic 43 (3):487-491.
  20.  8
    A Jump Operator in Set Recursion.Dag Normann - 1979 - Mathematical Logic Quarterly 25 (13‐18):251-264.
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  21.  30
    A Jump Operator in Set Recursion.Dag Normann - 1979 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 25 (13-18):251-264.
  22.  6
    A Jump Operator in Set Recursion.Dag Normann - 1982 - Journal of Symbolic Logic 47 (4):902-902.
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  23.  22
    Akira Kanda. Recursion theorems and effective domains. Annals of pure and applied logic, vol. 38 , pp. 289–300.Dag Normann - 1991 - Journal of Symbolic Logic 56 (1):335.
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  24.  5
    Aspects of the Continuous Functionals.Dag Normann - 1985 - In Anil Nerode & Richard A. Shore (eds.), Recursion Theory. American Mathematical Society. pp. 42--171.
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  25.  12
    Continuity, proof systems and the theory of transfinite computations.Dag Normann - 2002 - Archive for Mathematical Logic 41 (8):765-788.
    We use the theory of domains with totality to construct some logics generalizing ω-logic and β-logic and we prove a completenes theorem for these logics. The key application is E-logic, the logic related to the functional 3E. We prove a compactness theorem for sets of sentences semicomputable in 3E.
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  26.  14
    Characterizing the continuous functionals.Dag Normann - 1983 - Journal of Symbolic Logic 48 (4):965-969.
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  27.  20
    David Marker, Lectures on Infinitary Model Theory: Series: Lecture Notes in Logic, Vol. 46, 2016, pp. 192. ISBN-13: 978-1107181939 $118.00, ISBN-10: 1107181933 $107.08.Dag Normann - 2018 - Studia Logica 106 (6):1319-1323.
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  28.  8
    Degrees of functionals.Dag Normann - 1979 - Annals of Mathematical Logic 16 (3):269.
  29.  12
    Experiments on an Internal Approach to Typed Algorithms in Analysis.Dag Normann - 2011 - In S. B. Cooper & Andrea Sorbi (eds.), Computability in Context: Computation and Logic in the Real World. World Scientific. pp. 297.
  30.  14
    General type-structures of continuous and countable functionals.Dag Normann - 1983 - Mathematical Logic Quarterly 29 (4):177-192.
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  31.  10
    Hereditarily effective typestreams.Dag Normann - 1997 - Archive for Mathematical Logic 36 (3):219-225.
    We prove that the hierarchy of hereditarily effective typestreams, that are effective models of inductivly defined types, has the length of the first recursivly inaccessible ordinal.
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  32.  13
    Hyperfinite type structures.Dag Normann, Erik Palmgren & Viggo Stoltenberg-Hansen - 1999 - Journal of Symbolic Logic 64 (3):1216-1242.
  33. Hyperfinite Type Structures.Dag Normann, Erik Palmgren & Viggo Stoltenberg-Hansen - 1999 - Journal of Symbolic Logic 64 (3):1216-1242.
     
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  34.  25
    Limit spaces and transfinite types.Dag Normann & Geir Waagb - 2002 - Archive for Mathematical Logic 41 (6):525-539.
    We give a characterisation of an extension of the Kleene-Kreisel continuous functionals to objects of transfinite types using limit spaces of transfinite types.
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  35.  13
    Lawrence S. Moss. Power set recursion. Annals of pure and applied logic, vol. 71 , pp. 247–306.Dag Normann - 1996 - Journal of Symbolic Logic 61 (4):1388-1389.
  36.  40
    On abstract 1-sections.Dag Normann - 1974 - Synthese 27 (1-2):259 - 263.
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  37.  2
    On Robust Theorems Due to Bolzano, Weierstrass, Jordan, and Cantor.Dag Normann & Sam Sanders - forthcoming - Journal of Symbolic Logic:1-51.
    Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics where the aim is to identify theminimalaxioms needed to prove a given theorem from ordinary, i.e., non-set theoretic, mathematics. This program has unveiled surprising regularities: the minimal axioms are very oftenequivalentto the theorem over thebase theory, a weak system of ‘computable mathematics’, while most theorems are either provable in this base theory, or equivalent to one of onlyfourlogical systems. The latter plus the base theory are called the ‘Big (...)
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  38.  1
    R.E. degrees of continuous functionals.Dag Normann - 1983 - Archive for Mathematical Logic 23 (1):79-98.
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  39.  16
    Representation theorems for transfinite computability and definability.Dag Normann - 2002 - Archive for Mathematical Logic 41 (8):721-741.
    We show how Kreisel's representation theorem for sets in the analytical hierarchy can be generalized to sets defined by positive induction and use this to estimate the complexity of constructions in the theory of domains with totality.
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  40.  1
    The Continuous Functionals as Limit Spaces.Dag Normann - 2012 - In Ulrich Berger, Hannes Diener, Peter Schuster & Monika Seisenberger (eds.), Logic, Construction, Computation. De Gruyter. pp. 353-380.
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  41.  7
    The Computational Power of ℳω.Dag Normann & Christian Rørdam - 2002 - Mathematical Logic Quarterly 48 (1):117-124.
    We prove that the Kleene schemes for primitive recursion relative to the μ-operator, relativized to some nondeterministic objects, have the same power to express total functionals when interpreted over the partial continuous functionals and over the Kleene-Kreisel continuous functionals. Relating the former interpretation to Niggl's ℳω we prove Nigg's conjecture that ℳω is strictly weaker than Plotkin's PCF + PA.
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  42.  8
    The Computational Power of ℳ.Dag Normann & Christian Rørdam - 2002 - Mathematical Logic Quarterly 48 (1):117-124.
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  43.  13
    The extensional ordering of the sequential functionals.Dag Normann & V. Yu Sazonov - 2012 - Annals of Pure and Applied Logic 163 (5):575-603.
  44.  5
    Closing the gap between the continuous functionals and recursion in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $^3E$\end{document}. [REVIEW]Dag Normann - 1997 - Archive for Mathematical Logic 36 (4-5):269-287.
    We show that the length of a hierarchy of domains with totality, based on the standard domain for the natural numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\Bbb N}$\end{document} and closed under dependent products of continuously parameterised families of domains will be the first ordinal not recursive in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $^3E$\end{document} and any real. As a part of the proof we show that the domains of the hierarchy share (...)
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  45.  19
    Arnold Beckmann and Wolfram Pohlers. Applications of cut-free infinitary derivations to generalized recursion theory. Annals of pure and applied logic, vol. 94 , pp. 7–19. [REVIEW]Dag Normann - 2000 - Bulletin of Symbolic Logic 6 (2):221-222.
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  46.  8
    Gerald E. Sacks. Higher recursion theory. Perspectives in mathematical logic. Springer-Verlag, Berlin etc. 1990, xv + 344 pp. [REVIEW]Dag Normann - 1992 - Journal of Symbolic Logic 57 (2):761-762.
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  47.  12
    L. L. Ivanov. Algebraic recursion theory. Edited by J. L. Bell. Mathematics and its applications. Ellis Horwood Limited, Chichester, West Sussex, 1987 , also distributed by Halsted Press, John Wiley & Sons, New York etc., 256 pp. [REVIEW]Dag Normann - 1988 - Journal of Symbolic Logic 53 (3):986-987.
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  48.  10
    Review: Arnold Beckmann, Wolfram Pohlers, Applications of Cut-Free Infinitary Derivations to Generalized Recursion Theory. [REVIEW]Dag Normann - 2000 - Bulletin of Symbolic Logic 6 (2):221-222.
  49.  16
    Review: Akira Kanda, Recursion Theorems and Effective Domains. [REVIEW]Dag Normann - 1991 - Journal of Symbolic Logic 56 (1):335-335.
  50.  8
    Review: Dimiter G. Skordev, Computability in Combinatory Spaces. An Algebraic Generalization of Abstract First Order Computability. [REVIEW]Dag Normann - 1995 - Journal of Symbolic Logic 60 (2):695-696.
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