A Candidate for the Generalised Real Line

  • Lorenzo Galeotti
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9709)


Let \(\kappa \) be an uncountable cardinal with \(\kappa ^{<\kappa }=\kappa \). In this paper we introduce \({{\mathrm{\mathbb {R}}}}_\kappa \), a Cauchy-complete real closed field of cardinality \(2^\kappa \). We will prove that \({{\mathrm{\mathbb {R}}}}_\kappa \) shares many features with \({{\mathrm{\mathbb {R}}}}\) which have a key role in real analysis and computable analysis. In particular, we will prove that the Intermediate Value Theorem holds for a non-trivial subclass of continuous functions over \({{\mathrm{\mathbb {R}}}}_\kappa \). We propose \({{\mathrm{\mathbb {R}}}}_\kappa \) as a candidate for extending computable analysis to generalised Baire spaces.


  1. 1.
    Alling, N.: Foundations of Analysis over Surreal Number Fields. North-Holland Mathematics Studies, Elsevier Science, Holland (1987)zbMATHGoogle Scholar
  2. 2.
    Asperó, D., Tsaprounis, K.: Long reals (2015).
  3. 3.
    Brattka, V.: Computable versions of Baire’s category theorem. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 224–235. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Brattka, V., Gherardi, G.: Effective choice and boundedness principles in computable analysis. Bull. Symbolic Logic 17(1), 73–117 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cohn, P.: Basic Algebra: Groups, Rings, and Fields. Springer, Heidelberg (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Conway, J.: On Numbers and Games. Ak Peters Series. Taylor & Francis, London (2000)zbMATHGoogle Scholar
  7. 7.
    Dales, H., Woodin, W.: Super-real Fields: Totally Ordered Fields with Additional Structure. London Mathematical Society Monographs. Clarendon Press, Oxford (1996)zbMATHGoogle Scholar
  8. 8.
    Dries, L., Ehrlich, P.: Fields of surreal numbers and exponentiation. Fundamenta Mathematicae 167, 173–188 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ehrlich, P.: Real Numbers, Generalizations of the Reals, and Theories of Continua. Synthese Library. Springer, Heidelberg (1994)CrossRefzbMATHGoogle Scholar
  10. 10.
    Ehrlich, P.: Dedekind cuts of Archimedean complete ordered abelian groups. Algebra universalis 37(2), 223–234 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ehrlich, P., Costin, O., Friedman, H.: Integration on the surreals: a conjecture of Conway, Kruskal and Norton (2015). arXiv:1505.02478
  12. 12.
    Friedman, S., Hyttinen, T., Kulikov, V.: Generalized Descriptive Set Theory and Classification Theory. Memoirs of the American Mathematical Society. American Mathematical Society, Providence (2014)zbMATHGoogle Scholar
  13. 13.
    Galeotti, L.: Computable Analysis Over the Generalized Baire Space. Master’s thesis, ILLC Master of Logic Thesis Series MoL-2015-13, Universiteit van Amsterdam (2015)Google Scholar
  14. 14.
    Gonshor, H.: An Introduction to the Theory of Surreal Numbers. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1986)CrossRefzbMATHGoogle Scholar
  15. 15.
    Klaua, D.: Rational and real ordinal numbers. In: Ehrlich, P. (ed.) Real Numbers, Generalizations of the Reals, and Theories of Continua. Synthese Library, vol. 242, pp. 259–276. Springer, Amsterdam (1994)CrossRefGoogle Scholar
  16. 16.
    Koepke, P.: Turing computations on ordinals. Bull. Symbolic Logic 11(3), 377–397 (2005)CrossRefzbMATHGoogle Scholar
  17. 17.
    Marker, D.: Model Theory: An Introduction. Graduate Texts in Mathematics. Springer, New York (2006)zbMATHGoogle Scholar
  18. 18.
    Munkres, J.: Topology. Featured Titles for Topology Series. Prentice Hall, Upper Saddle River (2000)Google Scholar
  19. 19.
    Rubinstein-Salzedo, S., Swaminathan, A.: Analysis on surreal numbers. J. Logic Anal. 6, 1–39 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sikorski, R.: On an ordered algebraic field. Soc. Sci. Litt. Varsovic Sci. Math. Phys. 41, 69–96 (1948)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Weihrauch, K.: Computable Analysis: An Introduction. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2012)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Universität HamburgHamburgGermany

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