The Notion of Forcing

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

Abstract

In this chapter we present a general technique, called forcing, for extending models of ZFC. The main ingredients to construct such an extension are a model V of ZFC (e.g., V=L), a partially ordered set ℙ=(P,≤) contained in V, as well as a special subset G of P which will not belong to V. The extended model V[G] will then consist of all sets which can be “described” or “named” in V, where the “naming” depends on the set G. The main task will be to prove that V[G] is a model of ZFC as well as to decide (within V) whether a given statement is true or false in a certain extension V[G].

To get an idea of how this is done, think for a moment that there are people living in V. Notice that for these people, V is the unique set-theoretic universe which contains all sets. Now, the key point is that for any statement, these people are in fact able to compute whether the statement is true or false in a particular extension V[G], even though they have almost no information about the set G (in fact, they would actually deny the existence of such a set).

References

  1. 1.
    Paul J. Cohen: The independence of the continuum hypothesis I. Proc. Natl. Acad. Sci. USA 50, 1143–1148 (1963) CrossRefGoogle Scholar
  2. 2.
    Paul J. Cohen: The independence of the continuum hypothesis II. Proc. Natl. Acad. Sci. USA 51, 105–110 (1964) CrossRefGoogle Scholar
  3. 3.
    Paul J. Cohen: Independence results in set theory. In: The Theory of Models, Proceedings of the 1963 International Symposium at Berkeley, J.W. Addison, L. Henkin, A. Tarski (eds.). Studies in Logic and the Foundation of Mathematics, pp. 39–54. North-Holland, Amsterdam (1965) Google Scholar
  4. 4.
    Paul J. Cohen: Set Theory and the Continuum Hypothesis. Benjamin, New York (1966) MATHGoogle Scholar
  5. 5.
    Paul J. Cohen: The discovery of forcing, Proceedings of the Second Honolulu Conference on Abelian Groups and Modules, Honolulu, HI, 2001, vol. 32, pp. 1071–1100 (2002) Google Scholar
  6. 6.
    Solomon Feferman: Some applications of the notions of forcing and generic sets. Fundam. Math. 56, 325–345 (1964/1965) MathSciNetGoogle Scholar
  7. 7.
    Akihiro Kanamori: Cohen and set theory. Bull. Symb. Log. 14, 351–378 (2008) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Kenneth Kunen: Set Theory, an Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics, vol. 102. North-Holland, Amsterdam (1983) MATHGoogle Scholar
  9. 9.
    Gregory H. Moore: The origins of forcing. In: Logic Colloquium ’86, Proceedings of the Colloquium held in Hull, U.K., July 13–19, 1986, F.R. Drake, J.K. Truss (eds.). Studies in Logic and the Foundation of Mathematics, vol. 124, pp. 143–173. North-Holland, Amsterdam (1988) Google Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZürichSwitzerland

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