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Proofs of Impossibility

  • Pavel Pudlák
Chapter
  • 2.5k Downloads
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

This chapter is about results that show the impossibility to derive some theorems from particular sets of axioms and the impossibility to decide some problems using algorithms. In order to put the things into a historical context, we start with classical results from geometry and algebra. We explain Gödel’s incompleteness theorems and sketch their proofs. We show the algorithmic undecidability of the halting problem. We present concrete theorems that are independent of the axioms of Peano Arithmetic and some stronger theories. In the last section, we explain the method of forcing, which is used to prove independence results in set theory.

Keywords

Peano Arithmetic Incompleteness Theorem Goodstein Sequence Paris-Harrington Theorem Finite Ramsey Theorem 
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.

References

  1. 10.
    Avigad, J., Sommer, R.: A model-theoretical approach to ordinal analysis. Bull. Symb. Log. 3, 17–59 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 21.
    Berger, R.: The undecidability of the domino problem. Mem. Am. Math. Soc. 66 (1966) Google Scholar
  3. 38.
    Carnap, R.: Logische Syntax der Sprache. Springer, Berlin (1934) zbMATHGoogle Scholar
  4. 46.
    Cohen, P.: Set Theory and the Continuum Hypothesis, Benjamin, New York (1963) Google Scholar
  5. 47.
    Conway J.H: Unpredictable iterations. In: Proc. 1972 Number Th. Conf, pp. 49–52. University Press of Colorado, Boulder (1972) Google Scholar
  6. 57.
    Davis, M., Putnam, H., Robinson, J.: The decision problem for exponential Diophantine equations. Ann. Math. (2) 74(3), 425–436 (1961) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 80.
    Friedberg, R.M.: Two recursively enumerable sets of incomparable degrees of unsolvability. Proc. Natl. Acad. Sci. USA 43, 236–238 (1957) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 82.
    Friedman, H.: Finite functions and the necessary use of large cardinals. Ann. Math. 148, 803–893 (1998) zbMATHCrossRefGoogle Scholar
  9. 97.
    Gödel, K.: The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. Annals of Mathematical Studies, vol. 3. Princeton University Press, Princeton (1940) Google Scholar
  10. 105.
    Goodstein, R.L.: On the restricted ordinal theorem. J. Symb. Log. 9, 33–41 (1944) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 125.
    Hilbert, D.: Mathematical problems. Bull. Am. Math. Soc. 8, 437–479 (1902) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 141.
    Jockusch, C.G., Jr.: Ramsey’s theorem and recursion theory. J. Symb. Log. 37, 268–280 (1972) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 143.
    Jones, J.P.: Universal Diophantine equation. J. Symb. Log. 47, 549–571 (1982) zbMATHCrossRefGoogle Scholar
  14. 146.
    Kahr, A.S., Moore, E.F., Wang, H.: Entscheidungsproblem reduced to the AEA case. Proc. Natl. Acad. Sci. USA 48(3), 365–377 (1962) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 149.
    Kanamori, A., McAloon, K.: On Gödel incompleteness and finite combinatorics. Ann. Pure Appl. Log. 33(1), 23–41 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 150.
    Katz, V.J.: A History of Mathematics – an Introduction. Harper Collins, New York (1993) zbMATHGoogle Scholar
  17. 151.
    Ketonen, J., Solovay, R.: Rapidly growing Ramsey functions. Ann. Math. 113, 267–314 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 152.
    Kirby, L., Paris, J.: Accessible independence results for Peano arithmetic. Bull. Lond. Math. Soc. 14, 285–293 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 155.
    Kleene, S.C.: Extension of an effectively generated class of functions by enumeration. Colloq. Math. 6(1), 67–78 (1958) MathSciNetzbMATHGoogle Scholar
  20. 171.
    Kruskal, J.B.: Well-quasi-ordering, the tree theorem, and Vázsonyi’s conjecture. Trans. Am. Math. Soc. 95, 210–225 (1960) MathSciNetzbMATHGoogle Scholar
  21. 192.
    Martin, D.A., Solovay, R.M.: Internal Cohen extensions. Ann. Math. Log. 2(2), 143–178 (1970) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 193.
    Matiyasevich Yu, V.: Enumerable sets are Diophantine. Dokl. Akad. Nauk SSSR 191(2), 279–282 (1970). (In Russian; English translation: Sov. Math. Dokl. 11(2), 354–358) MathSciNetGoogle Scholar
  23. 201.
    Mostowski, A.: A generalization of the incompleteness theorem. Fundam. Math. 49, 205–232 (1961) MathSciNetzbMATHGoogle Scholar
  24. 202.
    Muchnik, A.A.: On the unsolvability of the problem of reducibility in the theory of algorithms. Dokl. Akad. Nauk SSSR 108, 194–197 (1956) (Russian) MathSciNetzbMATHGoogle Scholar
  25. 212.
    Paris, J.B.: Some independence results for Peano arithmetic. J. Symb. Log. 43(4), 725–731 (1978) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 213.
    Paris, J.B.: A hierarchy of cuts in models of arithmetic. In: Model Theory of Algebra and Arithmetic, Karpacz, 1979. Springer Lecture Notes in Math., vol. 834, pp. 312–337 (1980) CrossRefGoogle Scholar
  27. 214.
    Paris, J., Harrington, L.: A mathematical incompleteness in Peano arithmetic. In: Barwise, J. (ed.) Handbook of Mathematical Logic, pp. 1133–1142. North-Holland, Amsterdam (1977) CrossRefGoogle Scholar
  28. 218.
    Planck, M.: Where Is Science Going? Norton, New York (1932) Google Scholar
  29. 222.
    Pudlák, P.: Cuts, consistency statements and interpretations. J. Symb. Log. 50(2), 423–441 (1985) zbMATHCrossRefGoogle Scholar
  30. 249.
    Rosser, J.B.: Extensions of some theorems of Gödel and Church. J. Symb. Log. 1, 87–91 (1936) zbMATHCrossRefGoogle Scholar
  31. 262.
    Scott, D., Solovay, R.: Boolean-Valued Models for Set Theory. Proc. AMS Summer Institute on Set Theory, Los Angeles. University of California, Berkeley (1967) Google Scholar
  32. 269.
    Simpson, S.G.: Nonprovability of certain combinatorial properties of finite trees. In: Harrington, L.A., et al. (eds.) Harvey’s Friedman Research on the Foundations of Mathematics, pp. 87–117. North-Holland, Amsterdam (1985) CrossRefGoogle Scholar
  33. 274.
    Smoryński, C.: The varieties of arboreal experience. Math. Intell. 4, 182–188 (1982) zbMATHCrossRefGoogle Scholar
  34. 275.
    Smoryński, C.: Nonstandard models and related developments. In: Harrington, L.A., et al. (eds.) Harvey’s Friedman Research on the Foundations of Mathematics, pp. 179–229. North-Holland, Amsterdam (1985) CrossRefGoogle Scholar
  35. 280.
    Solovay, R.M.: Provability interpretations of modal logic. Isr. J. Math. 25, 287–304 (1976) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 287.
    Struik, D.J.: A Concise History of Mathematics. Dover, New York (1948) zbMATHGoogle Scholar
  37. 290.
    Tarski, A.: Der Wahrheitsbegriff in den formalisierten Sprachen. Stud. Philos. 1, 261–405 (1936) Google Scholar
  38. 293.
    Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. (2) 42, 230–265 2(1937) MathSciNetCrossRefGoogle Scholar
  39. 300.
    Vopěnka, P.: On ∇-model of set theory. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 13, 267–272 (1965) zbMATHGoogle Scholar
  40. 304.
    Wang, H.: Proving theorems by pattern recognition–II. Bell Syst. Tech. J. 40(1), 1–41 (1961) Google Scholar
  41. 306.
    Wang, H.: Some facts about Kurt Gödel. J. Symb. Log. 46(3), 653–659 (1981) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Pavel Pudlák
    • 1
  1. 1.ASCRPragueCzech Republic

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