Mathematician’s World

  • Pavel Pudlák
Part of the Springer Monographs in Mathematics book series (SMM)


We start by explaining which objects are studied in mathematics. We follow the standard approach in which these objects are mathematical structures. We discuss the crucial role of sets in the foundations of mathematics and briefly sketch how they are used to define some basic arithmetical structures. In this introductory chapter we state the basic axioms about sets informally as principles. We present the most important antinomies of intuitive set theory. We discuss the axiomatic method, which was discovered in antiquity and is the basis of all contemporary mathematical theories. Finally, we explain why abstract concepts are useful, even if we only need to solve a problem stated in elementary terms.


Natural Number Boolean Function Binary Relation Binary Operation Proof System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 3.
    Alekhnovich, M.: Mutilated chessboard problem is exponentially hard for resolution. Theor. Comput. Sci. 310(1–3), 513–525 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 5.
    Appel, K., Haken, W.: Every planar map is four colorable. Part I. Discharging. Ill. J. Math. 21, 429–490 (1977) MathSciNetzbMATHGoogle Scholar
  3. 6.
    Appel, K., Haken, W., Koch, J.: Every planar map is four colorable. Part II. Reducibility. Ill. J. Math. 21, 491–567 (1977) MathSciNetzbMATHGoogle Scholar
  4. 59.
    Dedekind, R.: Was sind und was sollen die Zahlen? 1. Auflage. Vieweg, Braunschweig (1888) Google Scholar
  5. 65.
    Ebbinghaus, H.-D., Peckhaus, V.: Ernst Zermelo: An Approach to His Life and Work. Springer, Berlin (2007) zbMATHGoogle Scholar
  6. 68.
    Erdős, P.: On a new method in elementary number theory which leads to an elementary proof of the prime number theorem. Proc. Natl. Acad. Sci. USA 35, 374–384 (1949) CrossRefGoogle Scholar
  7. 69.
    Erdős, P., Szekerés, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935) Google Scholar
  8. 112.
    Hales, T.C.: A proof of the Kepler conjecture. Ann. Math. (2) 162(3), 1065–1185 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 124.
    Hilbert, D.: Grundlagen der Geometrie. Teubner, Berlin (1899) Google Scholar
  10. 195.
    McCarthy, J.: A tough nut for proof procedures. Stanford Artificial Intelligence Project, Memo No. 16 (1964) Google Scholar
  11. 210.
    Odlyzko, A.M., te Riele, H.J.J.: Disproof of the Mertens conjecture. J. Reine Angew. Math. 357, 138–160 (1985) MathSciNetzbMATHGoogle Scholar
  12. 216.
    Peano, G.: Arithmetices Principia, Nova Methodo Exposita. Fratres Bocca, Torino (1889) Google Scholar
  13. 230.
    Pudlák, P., Rödl, V., Sgall, J.: Boolean circuits, tensor ranks and communication complexity. SIAM J. Comput. 26(3), 605–633 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 235.
    Ramsey, F.P.: On a problem of formal logic. Proc. Lond. Math. Soc. 30(1), 264–286 (1930) MathSciNetCrossRefGoogle Scholar
  15. 254.
    Russell, B.: The Autobiography of Bertrand Russell, vol. 1. Allen & Unwin, London (1967) Google Scholar
  16. 263.
    Selberg, A.: An elementary proof of the prime-number theorem. Ann. Math. (2) 50, 305–313 (1949) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 289.
    Tao, T.: Every odd number greater than 1 is the sum of at most five primes. Math. Comput. (to appear) Google Scholar
  18. 295.
    Uhlig, D.: On the synthesis of self-correcting schemes from functional elements with a small number of reliable elements. Mat. Zametki 15(6), 937–944 (1974) MathSciNetGoogle Scholar
  19. 298.
    Vaught, R.L.: Axiomatizability by a schema. J. Symb. Log. 32(4), 473–479 (1967) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 299.
    Vinogradov, I.M.: Representation of an odd number as a sum of three primes. C. R. Acad. Sci. USSR 15, 191–249 (1937) Google Scholar
  21. 318.
    Zermelo, E.: Untersuchungen über die Grundlagen der Mengenlehre. I. Math. Ann. 65, 261–281 (1908) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Pavel Pudlák
    • 1
  1. 1.ASCRPragueCzech Republic

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