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Kronecker’s density theorem and irrational numbers in constructive reverse mathematics

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Abstract

To prove Kronecker’s density theorem in Bishop-style constructive analysis one needs to define an irrational number as a real number that is bounded away from each rational number. In fact, once one understands “irrational” merely as “not rational”, then the theorem becomes equivalent to Markov’s principle. To see this we undertake a systematic classification, in the vein of constructive reverse mathematics, of logical combinations of “rational” and “irrational” as predicates of real numbers.

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References

  1. Aczel, P.: The type theoretic interpretation of constructive set theory. In: Macintyre, A., Pacholski, L., Paris, J. (eds.) Logic Colloquium ’77, pp. 55–66. North-Holland, Amsterdam (1978)

    Chapter  Google Scholar 

  2. Aczel, P.: The type theoretic interpretation of constructive set theory: choice principles. In: Troelstra, A.S., van Dalen, D. (eds.) The L.E.J. Brouwer Centenary Symposium, pp. 1–40. North-Holland, Amsterdam (1982)

    Chapter  Google Scholar 

  3. Aczel, P.: The type theoretic interpretation of constructive set theory: inductive definitions. In: Barcan Marcus, R., Dorn, G.J.W., Weingartner, P. (eds.) Logic, Methodology, and Philosophy of Science VII, pp. 17–49. North-Holland, Amsterdam (1986)

    Chapter  Google Scholar 

  4. Aczel, P., Rathjen, M.: Notes on Constructive Set Theory. Institut Mittag-Leffler Preprint No. 40 (2000/01)

  5. Beeson, M.: Foundations of Constructive Mathematics. Springer, Berlin and Heidelberg (1985)

    MATH  Google Scholar 

  6. Berger, J.: Constructive equivalents of the uniform continuity theorem. J. UCS 11, 1878–1883 (2005)

    MATH  MathSciNet  Google Scholar 

  7. Berger, J.: The weak König lemma and uniform continuity. J. Symb. Log. 73, 933–939 (2008)

    Article  MATH  Google Scholar 

  8. Berger, J., Bridges, D.: A bizarre property equivalent to the Π0 1-fan theorem. Log. J. IGPL 14, 867–871 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Berger, J., Bridges, D.: A fan-theoretic equivalent of the antithesis of Specker’s theorem. Indag. Math. (N.S.) 18, 195–202 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Berger, J., Bridges, D.: The anti-Specker property, a Heine-Borel property, and uniform continuity. Arch. Math. Logic 46, 583–592 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Berger, J., Bridges, D., Schuster, P.: The fan theorem and unique existence of maxima. J. Symb. Log. 71, 713–720 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Berger, J., Ishihara, H.: Brouwer’s fan theorem and unique existence in constructive analysis. Math. Log. Quart. 51, 369–373 (2005)

    MathSciNet  Google Scholar 

  13. Berger, J., Schuster, P.: Classifying Dini’s theorem. Notre Dame J. Formal Logic 47, 253–262 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Bishop, E.: Foundations of Constructive Analysis. McGraw-Hill, New York (1967)

    MATH  Google Scholar 

  15. Bishop, E., Bridges, D.: Constructive Analysis. Springer, Berlin and Heidelberg (1985)

    MATH  Google Scholar 

  16. Bridges, D.: Constructive mathematics: a foundation for computable analysis. Theor. Comput. Sci. 219, 95–109 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Bridges, D.: A weak constructive sequential compactness property and the fan theorem. Log. J. IGPL 13, 151–158 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Bridges, D., Richman, F.: Varieties of Constructive Mathematics. Cambridge University Press, Cambridge (1987)

    MATH  Google Scholar 

  19. Bridges, D., Richman, F., Schuster, P.: A weak countable choice principle. Proc. Amer. Math. Soc. 128, 2749–2752 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  20. Bridges, D., Schuster, P.: A simple constructive proof of Kronecker’s density theorem. Elem. Math. 61, 152–154 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Bridges, D., V^it~a, L.: Techniques of Constructive Analysis. Springer, New York (2006)

    MATH  Google Scholar 

  22. van Dalen, D.: Logic and Structure, 4th edn. Springer, Berlin and Heidelberg (2004)

  23. David, R., Nour, K., Raffalli, C.: Introduction ‘a la logique. Théorie de la démonstration, 2nd ed. Dunod, Paris (2004)

  24. Diener, H., Loeb, I.: Sequences of real functions on [0,1] in constructive reverse mathematics. Ann. Pure Appl. Logic 157, 50–61 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 4th edn. Clarendon Press, Oxford (1960)

  26. Hensel, K. (ed.): Leopold Kroneckers Werke, Bd. III, Halbbd. I. Teubner, Leipzig (1899), and Chelsea Publ. Co., New York (1968)

  27. Heyting, A.: Die formalen Regeln der intuitionistischen Logik, pp. 42–56. Sitz. ber. Preuß. Akad. Wiss. Berlin Phys.-Math. Kl. (1930)

  28. Heyting, A.: Intuitionism. An Introduction. North-Holland, Amsterdam (1956)

    MATH  Google Scholar 

  29. Ishihara, H.: An omniscience principle, the König lemma and the Hahn-Banach theorem. Z. Math. Logik Grundlag. Math. 36, 237–240 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  30. Ishihara, H.: Informal constructive reverse mathematics. S~urikaisekikenky~usho K~oky~uroku 1381, 108–117 (2004)

    Google Scholar 

  31. Ishihara, H.: Constructive reverse mathematics: compactness properties. In: Crosilla, L., Schuster, P. (eds.) From Sets and Types to Topology and Analysis, Oxford Logic Guides 48, pp. 245–267. Oxford University Press, Oxford (2005)

  32. Ishihara, H.: Reverse mathematics in Bishop’s constructive mathematics. Philos. Sci., Cahier Spécial 6, 43–59 (2006)

    Google Scholar 

  33. Ishihara, H., Mines, R.: Various continuity properties in constructive analysis. In: Schuster, P., Berger, U., Osswald, H. (eds.) Reuniting the Antipodes. Constructive and Nonstandard Views of the Continuum. Proceedings 1999 Venice Symposion, Synthese Library 306, pp. 103–110. Kluwer, Dordrecht (2001)

  34. Ishihara, H., Schuster, P.: Compactness under constructive scrutiny. Math. Log. Q. 50, 540–550 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  35. Khalouani, M., Labhalla, S., Lombardi, H.: Étude constructive de problèmes de topologie pour les réels irrationels. Math. Log. Quart. 45, 257–288 (1999)

    MATH  MathSciNet  Google Scholar 

  36. Kronecker, L.: Näherungsweise ganzzahlige Auflösung linearer Gleichungen. Monatsber. Königl. Preuß. Akad. Wiss. Berlin, 1179–1193 and 1271–1299 (1884)

  37. Loeb, I.: Equivalents of the (weak) fan theorem. Ann. Pure Appl. Logic 132, 51–66 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  38. Loeb, I.: Indecomposability of R and R\{0} in constructive reverse mathematics. Log. J. IGPL 16, 269–273 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  39. Mandelkern, M.: Limited omniscience and the Bolzano–Weierstraß principle. Bull. London Math. Soc. 20, 319–320 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  40. Mandelkern, M.: Constructive irrational space. Manus. Math. 60, 397–406 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  41. Mines, R., Ruitenburg, W., Richman, F.: A Course in Constructive Algebra. Springer, New York (1987)

    Google Scholar 

  42. Rathjen, M.: Choice principles in constructive and classical set theories. In: Chatzidakis, Z., Koepke, P., Pohlers, W. (eds.) Logic Colloquium ’02. Proceedings, Münster, 2002. Lect. Notes Logic 27, pp. 299–326. Assoc. Symbol. Logic, La Jolla (2006)

  43. Richman, F.: Intuitionism as generalization. Philos. Math. 5(3), 124–128 (1990)

    MATH  MathSciNet  Google Scholar 

  44. Richman, F.: The fundamental theorem of algebra: a constructive development without choice. Pacific J. Math. 196, 213–230 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  45. Richman, F.: Constructive mathematics without choice. In: Schuster, P., Berger, U., Osswald, H. (eds.) Reuniting the Antipodes. Constructive and Nonstandard Views of the Continuum, Proceedings 1999 Venice Symposion, Synthese Library 306, pp. 199–205. Kluwer, Dordrecht (2001)

  46. Richman, F.: Real numbers and other completions. Math. Log. Quart. 54, 98–108 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  47. Schuster, P.M.: A constructive look at generalised Cauchy reals. Math. Log. Quart. 46, 125–134 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  48. Schuster, P.: Countable choice as a questionable uniformity principle. Philos. Math. 12(3), 106–134 (2004)

    MATH  MathSciNet  Google Scholar 

  49. Schuster, P.: Logisch zwingende Teilprinzipien von ZFC. Log. Anal. (N.S.) 48, 301–310 (2005)

    MATH  MathSciNet  Google Scholar 

  50. Schuster, P.: What is continuity, constructively? J. UCS 11, 2076–2085 (2005)

    MATH  MathSciNet  Google Scholar 

  51. Schuster, P.: Unique solutions. Math. Log. Quart. 52, 534–539 (2006). Corrigendum: Math. Log. Quart. 53, 214 (2007)

  52. Schuster, P., Schwichtenberg, H.: Constructive solutions of continuous equations. In: Link, G. (ed.) One Hundred Years of Russell’s Paradox. International Conference in Logic and Philosophy. München, Germany, June 2–5, 2001, De Gruyter Series in Logic and Its Applications 6, pp. 227–245. De Gruyter, Berlin (2004)

  53. Simpson, S.G.: Subsystems of Second Order Arithmetic. Springer, Berlin and Heidelberg (1999)

    MATH  Google Scholar 

  54. Specker, E.: Nicht konstruktiv beweisbare Sätze der Analysis. J. Symbol. Log. 14, 145–158 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  55. Taschner, R.J.: Eine Ungleichung von van der Corput und Kemperman. Monatsh. Math. 91, 139–152 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  56. Troelstra, A.S., van Dalen, D.: Constructivism in Mathematics. Two volumes. North-Holland, Amsterdam (1988)

  57. Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory, 2nd edn. Cambridge University Press, Cambridge (2000)

  58. Veldman, W.: Brouwer’s fan theorem as an axiom and as a contrast to Kleene’s alternative, Preprint. Radboud University, Nijmegen, 2005

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Authors and Affiliations

  1. School of Information Science, Japan Advanced Institute of Science and Technology, Nomi, 923-1292, Ishikawa, Japan

    Hajime Ishihara

  2. Department of Pure Mathematics, University of Leeds, LS2 9JT, Leeds, UK

    Peter Schuster

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  1. Hajime Ishihara
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  2. Peter Schuster
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Correspondence to Peter Schuster.

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Ishihara, H., Schuster, P. Kronecker’s density theorem and irrational numbers in constructive reverse mathematics . Math. Semesterber. 57, 57–72 (2010). https://doi.org/10.1007/s00591-009-0062-x

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