The Typical Constructible Object

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


Baire Category is an important concept in mathematical analysis. It gives a notion of large set, hence a way of identifying the properties of typical objects. One of the most important applications of Baire Category is to provide a way of proving the existence of objects with specified properties without having to give an explicit construction, showing at the same time that these properties are prevalent. For instance it has been extensively used in mathematical analysis to better understand and separate classes of real functions such as analytic and smooth functions (see [9] for a wide range of applications of the Baire Category Theorem in analysis).


  1. 1.
    Brattka, V., Hendtlass, M., Kreuzer, A.P.: On the uniform computational content of the Baire category theorem (2015). CoRR arXiv:1510.01913
  2. 2.
    Bridges, D., Ishihara, H., Vîă, L.: A new constructive version of Baire’s theorem. Hokkaido Math. J. 35(1), 107–118 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brown, D.K., Simpson, S.G.: The Baire category theorem in weak subsystems of second-order arithmetic. J. Symb. Logic 58(2), 557–578 (1993)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Richard, M.: Friedberg: two recursively enumerable sets of incomparable degrees of unsolvability (solution of Post’s problem, 1944). Proc. Natl. Acad. Sci. U.S.A. 43(2), 236–238 (1957)CrossRefMATHGoogle Scholar
  5. 5.
    Hoyrup, M.: Irreversible computable functions. In: Mayr, E.W., Portier, N. (eds.) 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), STACS 2014, 5–8 March 2014, Lyon, France, LIPIcs, vol. 25, pp. 362–373. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2014)Google Scholar
  6. 6.
    Hoyrup, M.: Genericity of weakly computable objects. Theory of Computing Systems (2016, to appear)Google Scholar
  7. 7.
    Ingrassia, M.A.: P-genericity for recursively enumerable sets. Ph.D. thesis, University of Illinois at Urbana-Champaign (1981)Google Scholar
  8. 8.
    Carl, G.: Jockush: simple proofs of some theorems on high degrees. Can. J. Math. 29, 1072–1080 (1977)CrossRefMATHGoogle Scholar
  9. 9.
    Jones, S.H.: Applications of the Baire category theorem. Real Anal. Exch. 23(2), 363–394 (1999)MathSciNetMATHGoogle Scholar
  10. 10.
    Kleene, S.C., Emil, L.: Post: the upper semi-lattice of degrees of recursive unsolvability. Ann. Math. 59(3), 379–407 (1954)CrossRefMATHGoogle Scholar
  11. 11.
    Jack, H.: Lutz: category and measure in complexity classes. SIAM J. Comput. 19(6), 1100–1131 (1990)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lutz, J.H.: Effective fractal dimensions. Mathe. Logic Q. 51(1), 62–72 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Muchnik, A.A.: On the unsolvability of the problem of reducibility in the theory of algorithms. Dokl. Akad. Nauk SSSR 108, 194–197 (1956)MathSciNetMATHGoogle Scholar
  14. 14.
    Nies, A.: Computability and Randomness. Oxford Logic Guides. Oxford University Press, Oxford (2009)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.LORIA, Inria Nancy Grand EstVillers-lès-NancyFrance

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