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Sets, Logic, and Computation

  • John StillwellEmail author
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Part of the Undergraduate Texts in Mathematics book series (UTM)

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In the 19th century, perennial concerns about the role of infinity in mathematics were finally addressed by the development of set theory and formal logic. Set theory was proposed as a mathematical theory of infinity and formal logic was proposed as a mathematical theory of proof (partly to avoid the paradoxes that seem to arise when reasoning about infinity). In this chapter we discuss these two developments, whose interaction led to mind-bending consequences in the 20th century. Both set theory and logic throw completely new light on the question, “What is mathematics?” But they turn out to be double-edged swords.
  • Set theory brings remarkable clarity to the concept of infinity, but it shows infinity to be unexpectedly complicated–in fact, more complicated than set theory itself can describe.

  • Formal logic encompasses all known methods of proof, but at the same time it shows these methods to be incomplete. In particular, any reasonably strong system of logic cannot prove its own consistency.

  • Formal logic is the origin of the concept of computability, which gives a rigorous definition of an algorithmically solvable problem. However, some important problems turn out to be unsolvable.

It might be thought that the limits of formal proof are too remote to be of interest to ordinary mathematicians. But in the next chapter we will show how these limits are now being reached in one of the most down-to-earth fields of mathematics: combinatorics.

Keywords

Turing Machine Computable Function Order Type Continuum Hypothesis Large Cardinal 
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.

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References

  1. Baer, R. (1928). Zur Axiomatik der Kardinalarithmetik. Math. Zeit. 29, 381–396.zbMATHCrossRefMathSciNetGoogle Scholar
  2. Banach, S. and A. Tarski (1924). Sur la décomposition des ensembles de points en parties respectivement congruentes. Fund. Math. 6, 244–277.Google Scholar
  3. Boole, G. (1847). Mathematical Analysis of Logic. Reprinted by Basil Blackwell, London, 1948.Google Scholar
  4. Borel, E. (1898). Leçons sur la théorie des fonctions. Paris: Gauthier-Villars.zbMATHGoogle Scholar
  5. Chaitin, G. J. (1970). Computational complexity and Gödel’s incompleteness theorem. Notices Amer. Math. Soc. 17, 672.Google Scholar
  6. Church, A. (1936). An unsolvable problem in elementary number theory. Amer. J. Math. 58, 345–363.CrossRefMathSciNetGoogle Scholar
  7. Church, A. (1938). Review. J. Symb. Logic 3, 46.Google Scholar
  8. du Bois-Reymond, P. (1875). Über asymptotische Werte, infinitäre Approximationen und infinitäre Auflösung von Gleichungen. Math. Ann. 8, 363–414.CrossRefMathSciNetGoogle Scholar
  9. Harnack, A. (1885). Über den Inhalt von Punktmengen. Math. Ann. 25, 241–250.CrossRefMathSciNetGoogle Scholar
  10. Hausdorff, F. (1914). Grundzüge der Mengenlehre. Leipzig: Von Veit.zbMATHGoogle Scholar
  11. Hilbert, D. and P. Bernays (1936). Grundlagen der Mathematik I. Berlin: Springer.Google Scholar
  12. Kanamori, A. (1994). The Higher Infinite. Berlin: Springer-Verlag.zbMATHGoogle Scholar
  13. Kreisel, G. (1980). Kurt Gödel. Biog. Mem. Fellows Roy. Soc. 26, 149–224.Google Scholar
  14. Post, E. L. (1944). Recursively enumerable sets of positive integers and their decision problems. Bull. Amer. Math. Soc. 50, 284–316.zbMATHCrossRefMathSciNetGoogle Scholar
  15. Ramsey, F. P. (1929). On a problem of formal logic. Proc. Lond. Math. Soc. 30, 291–310.Google Scholar
  16. Shelah, S. (1984). Can you take Solovay’s inaccessible away? Israel J. Math. 48(1), 1–47.zbMATHCrossRefMathSciNetGoogle Scholar
  17. Stillwell, J. (1993). Classical Topology and Combinatorial Group Theory, 2nd ed. New York, NY: Springer-Verlag.zbMATHGoogle Scholar
  18. Ulam, S. (1930). Zur Masstheorie in der allgemeinen Mengenlehre. Fund. Math. 15, 140–150.Google Scholar
  19. van Dalen, D. and A. Monna (1972). Sets and Integration. An Outline of the Development. Groningen: Wolters-Noordhoff Publishing.zbMATHGoogle Scholar
  20. Wagon, S. (1985). The Banach-Tarski Paradox. Cambridge: Cambridge University Press. With a foreword by Jan Mycielski.zbMATHGoogle Scholar
  21. Woodin, W. H. (1999). The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. Berlin: Walter de Gruyter & Co.zbMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of San FranciscoSan FranciscoUSA

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