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Why Sets?

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Pillars of Computer Science

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4800))

Abstract

Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a far-away planet. Would their mathematics be set-based? What are the alternatives to the set-theoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.

This is a revised version of an article originally published in the Bulletin of the European Association for Theoretical Computer Science, Number 84, October 2004, and republished here with permission of the Association.

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References

  1. Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley, Reading (1995)

    MATH  Google Scholar 

  2. Ackermann, W.: Zur Axiomatik der Mengenlehre. Math. Ann. 131, 336–345 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  3. Aczel, P.: Non-Well-Founded Sets. CSLI Lecture Notes 14, Center for the Study of Language and Information. Stanford Univ. (1988)

    Google Scholar 

  4. Barendregt, H.: The Lambda Calculus. Its Syntax and Semantics. Studies in Logic and the Foundations of Mathematics, vol. 103. North-Holland, Amsterdam (1984)

    MATH  Google Scholar 

  5. Barendregt, H.: The impact of the lambda calculus in logic and computer science. Bull. Symbolic Logic 3, 181–215 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Barwise, J., Moss, L.: Vicious Circles. On the mathematics of non-wellfounded phenomena. CSLI Lecture Notes 60, Center for the Study of Language and Information. Stanford Univ. (1996)

    Google Scholar 

  7. Bernays, P.: A system of axiomatic set theory – Part I. J. Symbolic Logic 2, 65–77 (1937)

    Article  MATH  Google Scholar 

  8. Blass, A., Gurevich, Y., Shelah, S.: Choiceless polynomial time. Annals of Pure and Applied Logic 100, 141–187 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Blass, A., Gurevich, Y., Van den Bussche, J.: Abstract state machines and computationally complete query languages. Information and Computation 174, 20–36 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bourbaki, N.: Elements of Mathematics: Theory of Sets. Translation from French. Addison-Wesley, Reading (1968)

    MATH  Google Scholar 

  11. Chandra, A., Harel, D.: Structure and complexity of relational queries. J. Comput. and System Sciences 25, 99–128 (1982)

    Article  MATH  Google Scholar 

  12. Church, A.: A set of postulates for the foundation of logic. Ann. Math (2) 33, 346–366 (1933) and 34, 839–864 (1932)

    Google Scholar 

  13. Church, A.: An unsolvable problem of elementary number theory. Amer. J. Math. 58, 345–363 (1936)

    Article  MATH  MathSciNet  Google Scholar 

  14. Church, A.: A formulation of the simple theory of types. J. Symbolic Logic 5, 56–68 (1940)

    Article  MATH  MathSciNet  Google Scholar 

  15. Church, A.: The Calculi of Lambda-Conversion. Annals of Mathematics Studies, vol. 6. Princeton Univ. Press, Princeton (1941)

    Google Scholar 

  16. Curry, H.: Grundlagen der kombinatorischen Logik. Amer. J. Math. 52, 509–536, and 789–834 (1930)

    Google Scholar 

  17. Curry, H.: The combinatory foundations of mathematical logic. J. Symbolic Logic 7, 49–64 (1942)

    Article  MATH  MathSciNet  Google Scholar 

  18. Curry, H., Feys, R.: Combinatory Logic, vol. 1. North-Holland, Amsterdam (1958)

    MATH  Google Scholar 

  19. Deiser, O.: Orte, Listen, Aggregate. Habilitationsschrift, Freie Universität Berlin (2006)

    Google Scholar 

  20. Dijkstra, E.W.: Sets are unibags. Handwritten note EWD786 (April 1981), http://www.cs.utexas.edu/users/EWD/ewd07xx/EWD786a.PDF

  21. Enderton, H.: Elements of Set Theory. Academic Press, London (1977)

    MATH  Google Scholar 

  22. Feferman, S.: Set-theoretical foundations of category theory. In: Mac Lane, S. (ed.) Reports of the Midwest Category Seminar, III. Lecture Notes in Mathematics, vol. 106, pp. 201–247. Springer, Heidelberg (1969)

    Chapter  Google Scholar 

  23. Feferman, S.: A language and axioms for explicit mathematics. In: Crossley, J. (ed.) Algebra and Logic. Lecture Notes in Mathematics, vol. 450, pp. 87–139. Springer, Heidelberg (1975)

    Chapter  Google Scholar 

  24. Feferman, S.: Constructive theories of functions and classes. In: Boffa, M., van Dalen, D., McAloon, K. (eds.) Logic Colloquium 1978. Studies in Logic and the Foundations of Mathematics, vol. 97, pp. 159–224. North-Holland, Amsterdam (1980)

    Google Scholar 

  25. Feferman, S.: Toward useful type-free theories, I. J. Symbolic Logic 49, 75–111 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  26. Feferman, S., et al.: Does mathematics need new axioms. Bull. Symbolic Logic 6, 401–446 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. Fraenkel, A., Bar-Hillel, Y., Lévy, A.: Foundations of Set Theory. Studies in Logic and the Foundations of Mathematics, vol. 67. North-Holland, Amsterdam (1973)

    MATH  Google Scholar 

  28. Friedman, H.: Higher set theory and mathematical practice. Ann. Math. Logic 2, 325–357 (1970)

    Article  MathSciNet  Google Scholar 

  29. Gödel, K.: The consistency of the axiom of choice and of the generalized continuum hypothesis. Proc. Nat. Acad. Sci. U.S.A. 24, 556–557 (1938)

    Article  MATH  Google Scholar 

  30. Gödel, K.: The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis with the Axioms of Set Theory. Annals of Mathematics Studies, vol. 3. Princeton Univ. Press, Princeton (1940)

    Google Scholar 

  31. Gurevich, Y.: Logic and the challenge of computer science. In: Börger, E. (ed.) Current Trends in Theoretical Computer Science, pp. 1–57. Computer Science Press (1988)

    Google Scholar 

  32. Holmes, R.: New Foundations Home Page, http://math.boisestate.edu/~holmes/holmes/nf.html

  33. Isbell, J.: Review of [40]. Mathematical Reviews 34, 7332 (1967)

    Google Scholar 

  34. Jensen, R.: On the consistency of a slight(?) modification of Quine’s NF. Synthèse 19, 250–263 (1969)

    Article  Google Scholar 

  35. Johnstone, P.T.: Topos Theory. London Math. Soc. Monographs, vol. 10. Academic Press, London (1977)

    MATH  Google Scholar 

  36. Kanamori, A.: The empty set, the singleton, and the ordered pair. Bull. Symbolic Logic 9, 273–298 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  37. Kleene, S., Rosser, J.B.: The inconsistency of certain formal logics. Ann. Math. 36(2), 630–636 (1935)

    MathSciNet  Google Scholar 

  38. Lambek, J., Scott, P.: Introduction to Higher Order Categorical Logic. Cambridge Studies in Advanced Mathematics, vol. 7. Cambridge Univ. Press, Cambridge (1986)

    MATH  Google Scholar 

  39. Lawvere, F.W.: An elementary theory of the category of sets. Proc. Nat. Acad. Sci. U.S.A. 52, 1506–1511 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  40. Lawvere, F.W.: The category of categories as a foundation for mathematics. In: Proc. Conf. Categorical Algebra (La Jolla, CA, 1995), pp. 1–20. Springer, Heidelberg (1966)

    Google Scholar 

  41. Lawvere, F.W., Rosebrugh, R.: Sets for Mathematicians. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  42. Lawvere, F.W., Tierney, M.: Quantifiers and sheaves. In: Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp. 329–334. Gauthier-Villars (1971)

    Google Scholar 

  43. Lévy, A.: On Ackermann’s set theory. J. Symbolic Logic 24, 154–166 (1959)

    Article  MATH  Google Scholar 

  44. Mac Lane, S., Moerdijk, I.: Sheaves in geometry and logic. A first introduction to topos theory. Universitext. Springer, Heidelberg (1994)

    Google Scholar 

  45. Martin, D.A.: Borel determinacy. Ann. Math (2) 102(2), 363–371 (1975)

    Article  Google Scholar 

  46. Martin-Löf, P.: An intuitionistic theory of types: predicative part. In: Rose, H.E., Shepherdson, J.C. (eds.) Proceedings of the Logic Colloquium (Bristol, July, 1973). Studies in Logic and the Foundations of Mathematics, vol. 80, pp. 73–118. North-Holland, Amsterdam (1975)

    Google Scholar 

  47. von Neumann, J.: Eine Axiomatisierung der Mengenlehre. J. Reine Angew. Math. 154, 219–240 (1925) (English translation in From Frege To Gödel. In: van Heijenoort, J. (ed.) A Source Book in Mathematical Logic, 1879–1931, 393–413, Harvard University Press (1967))

    Google Scholar 

  48. von Neumann, J.: Die Axiomatisierung der Mengenlehre. Math. Z. 27, 669–752 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  49. Platek, R.: Eliminating the continuum hypothesis. J. Symbolic Logic 34, 219–225 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  50. Van Orman Quine, W.: New foundations for mathematical logic. Amer. Math. Monthly 44, 70–80 (1937)

    Article  MathSciNet  Google Scholar 

  51. Reinhardt, W.: Ackermann’s set theory equals ZF. Ann. Math. Logic 2, 189–249 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  52. Roitman, J.: The uses of set theory. Math. Intelligencer 14, 63–69 (1992)

    MATH  MathSciNet  Google Scholar 

  53. Rosser, J.B.: Logic for Mathematicians. McGraw-Hill, New York (1953)

    MATH  Google Scholar 

  54. Schönfinkel, M.: Über die Bausteine der mathematischen Logik. Math. Ann. 92, 305–316 (1924)

    Article  MathSciNet  MATH  Google Scholar 

  55. Schwartz, J.T., Dewar, R.B.K., Dubinsky, E., Schonberg, E.: Programming with Sets: An Introduction to SETL. Springer, Heidelberg (1986)

    MATH  Google Scholar 

  56. Shelah, S.: Can you take Solovay’s inaccessible away. Israel J. Math. 48, 1–47 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  57. Solovay, R.: A model of set theory in which every set of reals is Lebesgue measurable. Ann. Math. 92(2), 1–56 (1970)

    Article  MathSciNet  Google Scholar 

  58. Z users website, http://vl.zuser.org/

  59. Zermelo, E.: Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen 65, 261–281 (1908)

    Article  MathSciNet  MATH  Google Scholar 

  60. Zermelo, E.: Über Grenzzahlen und Mengenbereiche, Neue Untersuchungen über die Grundlagen der Mengenlehre. Fundamenta Mathematicae 16, 29–47 (1930)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, University of Michigan, Ann Arbor, MI 48109, USA

    Andreas Blass

  2. Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA

    Andreas Blass

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  1. Andreas Blass
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Arnon Avron Nachum Dershowitz Alexander Rabinovich

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Blass, A. (2008). Why Sets?. In: Avron, A., Dershowitz, N., Rabinovich, A. (eds) Pillars of Computer Science. Lecture Notes in Computer Science, vol 4800. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78127-1_11

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  • DOI: https://doi.org/10.1007/978-3-540-78127-1_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-78126-4

  • Online ISBN: 978-3-540-78127-1

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