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The Dedekind reals in abstract Stone duality
Published online by Cambridge University Press: 01 August 2009
Abstract
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Abstract Stone Duality (ASD) is a direct axiomatisation of general topology, in contrast to the traditional and all other contemporary approaches, which rely on a prior notion of discrete set, type or object of a topos.
ASD reconciles mathematical and computational viewpoints, providing an inherently computable calculus that does not sacrifice key properties of real analysis such as compactness of the closed interval. Previous theories of recursive analysis failed to do this because they were based on points; ASD succeeds because, like locale theory and formal topology, it is founded on the algebra of open subspaces.
ASD is presented as a lambda calculus, of which we provide a self-contained summary, as the foundational background has been investigated in earlier work.
The core of the paper constructs the real line using two-sided Dedekind cuts. We show that the closed interval is compact and overt, where these concepts are defined using quantifiers. Further topics, such as the Intermediate Value Theorem, are presented in a separate paper that builds on this one.
The interval domain plays an important foundational role. However, we see intervals as generalised Dedekind cuts, which underly the construction of the real line, not as sets or pairs of real numbers.
We make a thorough study of arithmetic, in which our operations are more complicated than Moore's, because we work constructively, and we also consider back-to-front (Kaucher) intervals.
Finally, we compare ASD with other systems of constructive and computable topology and analysis.
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References
Bauer, Andrej (2000). The Realizability Approach to Computable Analysis and Topology. PhD thesis, Carnegie Mellon University. Technical report CMU-CS-00-164.Google Scholar
Bauer, Andrej (2008). Efficient computation with Dedekind reals. In Fifth International Conference on Computability and Complexity in Analysis, Vasco, Brattka, Ruth, Dillhage, Tanja, Grubba, and Klutch, Angela, editors, Hagen, Germany.Google Scholar
Bauer, Andrej and Kavkler, Iztok (2008). Implementing real numbers with RZ. Electronic Notes in Theoretical Computer Science, 202:365–384.Google Scholar
Bauer, Andrej and Stone, Christopher (2008). RZ: a tool for bringing constructive and computable mathematics closer to programming practice. Journal of Logic and Computation, 19:17–43.Google Scholar
Berger, Josef and Bridges, Douglas (2008a). The anti-Specker property, a Heine–Borel property, and uniform continuity. Archive for Mathematical Logic, 46:583–592.CrossRefGoogle Scholar
Berger, Josef and Bridges, Douglas (2008b). The fan theorem and positive-valued uniformly continuous functions on compact intervals. New Zealand Journal of Mathematics, 38:129–135.Google Scholar
Bishop, Errett (1967). Foundations of Constructive Analysis. Higher Mathematics. McGraw–Hill.Google Scholar
Bishop, Errett and Bridges, Douglas (1985). Constructive Analysis. Number 279 in Grundlehren der mathematischen Wissenschaften. Springer-Verlag.Google Scholar
Bourbaki, Nicolas (1966). Topologie Générale. Hermann. English translation, General Topology, distributed by Springer-Verlag, 1989.Google Scholar
Bridges, Douglas (1999). Constructive mathematics: a foundation for computable analysis. Theoretical Computer Science, 219:95–109.Google Scholar
Bridges, Douglas and Richman, Fred (1987). Varieties of Constructive Mathematics. Number 97 in London Mathematical Society Lecture Notes. Cambridge University Press.CrossRefGoogle Scholar
Cederquist, Jan and Negri, Sara (1996). A constructive proof of the Heine–Borel covering theorem for formal reals. In Types for Proofs and Programs, Stefano, Beradi and Mario, Coppo, editors, number 1158 in Lecture Notes in Computer Science. Springer-Verlag.Google Scholar
Conway, John Horton (1976). On Numbers and Games. Number 6 in London Mathematical Society Monographs. Academic Press. Revised edition, 2001, published by A K Peters, Ltd.Google Scholar
Dedekind, Richard (1872). Stetigkeit und irrationale Zahlen. Braunschweig. Reprinted in (Dedekind 1932), pages 315–334; English translation, Continuity and Irrational Numbers, in (Dedekind 1901).Google Scholar
Dedekind, Richard (1901). Essays on the theory of numbers. Open Court. English translations by Beman, Wooster Woodruff; republished by Dover, 1963.Google Scholar
Dedekind, Richard (1932). Gesammelte mathematische Werke, volume 3. Vieweg, Braunschweig. Edited by Fricke, Robert, Noether, Emmy and Ore, Øystein; republished by Chelsea, New York, 1969.Google Scholar
Edalat, Abbas and Sünderhauf, Philipp (1998). A domain-theoretic approach to real number computation. Theoretical Computer Science, 210:73–98.CrossRefGoogle Scholar
Escardó, Martín (2004). Synthetic topology of data types and classical spaces. Electronic Notes in Theoretical Computer Science, 87:21–156.Google Scholar
Fourman, Michael and Hyland, Martin (1979). Sheaf models for analysis. In Applications of Sheaves, Michael, Fourman, Chris, Mulvey, and Scott, Dana, editors, number 753 in Lecture Notes in Mathematics. Springer-Verlag.CrossRefGoogle Scholar
Friedberg, Richard (1958). Un contre-example relatif aux fonctionnelles récursives. Comptes rendus hebdomadaires des scéances de l'Académie des Sciences (Paris), 247:852–854.Google Scholar
Gentzen, Gerhard (1935). Untersuchungen über das logische Schliessen. Mathematische Zeitschrift, 39:176–210, 405–431. English translation in (Gentzen 1969), pages 68–131.Google Scholar
Gentzen, Gerhard (1969). The Collected Papers of Gerhard Gentzen. Studies in Logic and the Foundations of Mathematics. North-Holland. Edited by Manfred, Szabo.Google Scholar
Gierz, Gerhard, Hoffmann, Karl Heinrich, Keimel, Klaus, Lawson, Jimmie, Mislove, Michael, and Scott, Dana (1980). A Compendium of Continuous Lattices. Springer-Verlag. Second edition, Continuous Lattices and Domains, published by Cambridge University Press, 2003.Google Scholar
Hanrot, Guillaume, Lefèvre, Vincent, Pélissier, Patrick, and Zimmermann, Paul. The MPFR Library. INRIA. www.mpfr.org.Google Scholar
Heyting, Arend (1956). Intuitionism, an Introduction. Studies in Logic and the Foundations of Mathematics. North-Holland. Third edition, 1971.Google Scholar
Hyland, Martin (1982). The effective topos. In The L.E.J. Brouwer Centenary Symposium, Anne, Troelstra and van Dalen, Dirk, editors. North Holland.Google Scholar
Hyland, Martin (1991). First steps in synthetic domain theory. In Category Theory, Aurelio, Carboni, Maria-Cristina, Pedicchio, and Rosolini, Giuseppe, editors, number 1488 in Lecture Notes in Mathematics. Springer Verlag.Google Scholar
Hyland, Martin and Rosolini, Giuseppe (1990). The discrete objects in the effective topos. Proceedings of the London Mathematical Society, 60:1–36.Google Scholar
Johnstone, Peter (1977). Topos Theory. Number 10 in London Mathematical Society Monographs. Academic Press.Google Scholar
Johnstone, Peter (1982). Stone Spaces. Number 3 in Cambridge Studies in Advanced Mathematics. Cambridge University Press.Google Scholar
Johnstone, Peter (1984). Open locales and exponentiation. Contemporary Mathematics, 30:84–116.Google Scholar
Joyal, André and Tierney, Myles (1984). An Extension of the Galois Theory of Grothendieck, volume 309 of Memoirs. American Mathematical Society.CrossRefGoogle Scholar
Julian, William and Richman, Fred (1984). A uniformly continuous function on [0,1] that is everywhere different from its infimum. Pacific Journal of Mathematics, 111:333–340.Google Scholar
Kaucher, Edgar (1980). Interval analysis in the extended interval space IR. In Fundamentals of Numerical Computation, Götz, Alefeld and Grigorieff, Rolf, editors, volume 2 of Computing. Supplementum. Springer-Verlag.Google Scholar
Kearfott, Baker (1996). Interval computations: Introduction, uses and resources. Euromath Bulletin, 2 (1):95–112.Google Scholar
Kelly, Max (1986). A survey of totality in ordinary and enriched categories. Cahiers de Géometrie et Topologie Differentielle, 27:109–132.Google Scholar
Kleene, Stephen (1945). On the interpretation of intuitionistic number theory. Journal of Symbolic Logic, 10:109–124.CrossRefGoogle Scholar
Kock, Anders (2006). Synthetic Differential Geometry. Number 333 in London Mathematical Society Lecture Note Series. Cambridge University Press, second edition.CrossRefGoogle Scholar
Kreinovich, Vladik, Nesterov, Vyacheslav, and Zheludeva, Nina (1996). Interval methods that are guaranteed to underestimate (and the resulting new justification of Kaucher arithmetic). Reliable Computing, 2 (2):119–124.CrossRefGoogle Scholar
Lakeyev, Anatoly (1995). Linear algebraic equations in Kaucher arithmetic. In Applications of Interval Computations (APIC'95), Vladik, Kreinovich, editor. supplement to Reliable Computing.Google Scholar
Lambov, Branimir (2007). RealLib: An efficient implementation of exact real arithmetic. Mathematical Structures in Computer Science, 17 (1):81–98. www.brics.dk/~barnie/RealLib/.Google Scholar
Müller, Norbert (2001). The iRRAM: Exact arithmetic in C++. In Computability and Complexity in Analysis: 4th International Workshop, CCA 2000 Swansea, UK, September 17, 2000, Selected Papers, Jens, Blanck, Vasco, Brattka, and Hertling, Peter, editors, number 2064 in Lecture Notes in Computer Science. Springer-Verlag.Google Scholar
Palmgren, Eric (2005). Continuity on the real line and in formal spaces. In From Sets and Types to Topology and Analysis: Towards Practicable Foundations of Constructive Mathematics, Laura, Crosilla and Schuster, Peter, editors, Oxford Logic Guides. Oxford University Press.Google Scholar
Peano, Giuseppe (1897). Studii di logica matematica. Atti della Reale Accademia di Torino, 32:565–583. Reprinted in Peano, Opere Scelte, Cremonese, 1953, vol. 2, pp. 201–217, and (in English) in Kennedy, Hubert, Selected Works of Giuseppe Peano, Toronto University Press, 1973, pp 190–205.Google Scholar
Plotkin, Gordon (1977). LCF considered as a programming language. Theoretical Computer Science, 5:223–255.CrossRefGoogle Scholar
Robinson, Abraham (1966). Non-standard Analysis. North-Holland. Revised edition, 1996, published by Princeton University Press.Google Scholar
Rogers, Hartley (1992). Theory of Recursive Functions and Effective Computability. MIT Press, third edition.Google Scholar
Rosemeier, Frank (2001). A constructive approach to Conway's theory of games. In Reuniting the Antipodes: Constructive and Nonstandard Views of the Continuum, Schuster, Peter, Berger, Ulrich, and Osswald, Horst, editors. Springer-Verlag.Google Scholar
Rosolini, Giuseppe (1986). Continuity and Effectiveness in Topoi. PhD thesis, University of Oxford.Google Scholar
Schuster, Peter (2003). Unique existence, approximate solutions, and countable choice. Theoretical Computer Science, 305:433–455.CrossRefGoogle Scholar
Schuster, Peter (2005). What is continuity, constructively? Journal of Universal Computer Science, 11 (12):2076–2085.Google Scholar
Scott, Dana (1972a). Continuous lattices. In Toposes, Algebraic Geometry and Logic, Bill, Lawvere, editor, number 274 in Lecture Notes in Mathematics. Springer-Verlag.Google Scholar
Scott, Dana (1972b). Lattice theory, data types and semantics. In Formal Semantics of Programming Languages, Rustin, Randall, editor. Prentice-Hall.Google Scholar
Stevenson, David (1985). Binary floating-point arithmetic. ANSI/IEEE Standard, 754. Revised, 2008.Google Scholar
Stolz, Otto (1883). Zur Geometrie der Alten, insbesondere über ein Axiom des Archimedes. Mathematische Annalen, 22 (4):504–519.Google Scholar
Street, Ross (1980). Cosmoi of internal categories. Transactions of the American Mathematical Society, 258:271–318.Google Scholar
Street, Ross and Walters, Robert (1978). Yoneda structures on 2-categories. Journal of Algebra, 50:350–379.Google Scholar
Taylor, Paul (1991). The fixed point property in synthetic domain theory. In Logic in Computer Science 6, Kahn, Gilles, editor. IEEE.Google Scholar
Taylor, Paul (1999). Practical Foundations of Mathematics. Number 59 in Cambridge Studies in Advanced Mathematics. Cambridge University Press.Google Scholar
[O]Taylor, Paul (2009) Foundations for Computable Topology. In Foundational Theories of Mathematics, Sommaruga, Giovanni, editor, Kluwer.Google Scholar
[A]Taylor, Paul (2002) Sober spaces and continuations. Theory and Applications of Categories, 10:248–299.Google Scholar
[B]Taylor, Paul (2002) Subspaces in abstract Stone duality. Theory and Applications of Categories, 10:300–366.Google Scholar
[C]Taylor, Paul (2000) Geometric and higher order logic using abstract Stone duality. Theory and Applications of Categories, 7:284–338.Google Scholar
[D]Taylor, Paul (2000) Non-Artin gluing in recursion theory and lifting in abstract Stone duality.Google Scholar
[E]Taylor, Paul (2005) Inside every model of Abstract Stone Duality lies an Arithmetic Universe. Electronic Notes in Theoretical Computer Science 122:247–296, Elsevier.Google Scholar
[G]Taylor, Paul (2006) Computably based locally compact spaces. Logical Methods in Computer Science, 2:1–70.Google Scholar
[H]Taylor, Paul (2004) An elementary theory of various categories of spaces and locales.Google Scholar
[J]Taylor, Paul (2005) A lambda calculus for real analysis. In Computability and Complexity in Analysis, Tanja, Grubba et al. , editors, volume 326 of Informatik Berichte, FernUniversität in Hagen.Google Scholar
[K]Taylor, Paul (2006) Interval analysis without intervals. In Real Numbers and Computers, Guillaume, Hanrot and Zimmermann, Paul, editors, Nancy.Google Scholar
Thielecke, Hayo (1997). Categorical Structure of Continuation Passing Style. PhD thesis, University of Edinburgh. ECS-LFCS-97-376.Google Scholar
Tholen, Walter (1980). A note on total categories. Bulletin of the Australian Mathematical Society, 21:169–173.Google Scholar
Troelstra, Anne Sjerp and van Dalen, Dirk (1988). Constructivism in Mathematics, an Introduction. Numbers 121 and 123 in Studies in Logic and the Foundations of Mathematics. North-Holland.Google Scholar
Vermeulen, Japie (1994). Proper maps of locales. Journal of Pure and Applied Algebra, 92:79–107.Google Scholar
Waaldijk, Frank (2005). On the foundations of constructive mathematics – especially in relation to the theory of continuous functions. Foundations of Science, 10 (3):249–324.Google Scholar
Wood, Richard (1982). Some remarks on total categories. Journal of Algebra, 75:538–545.Google Scholar
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