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Putnam’s Theorem on the Complexity of Models

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Book cover Hilary Putnam on Logic and Mathematics

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 9))

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Abstract

A streamlined proof of a theorem of Putnam’s: any satisfiable schema of predicate calculus has a model in which the predicates are interpreted as Boolean combinations of recursively enumerable relations. Related open problems are canvassed.

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Notes

  1. 1.

    I am grateful to Gary Ebbs and Carl G. Jockusch, Jr. for helpful comments.

References

  • Church, A. (1956). Introduction to mathematical logic. Princeton: Princeton University Press.

    Google Scholar 

  • Ebbs, G., & Goldfarb, W. (2018). First-order logical validity and the Hilbert-Bernays theorem. Philosophical Issues 28: Philosophy of Logic and Inference (forthcoming)

    Article  Google Scholar 

  • Enderton, H. (1972). A mathematical introduction to logic. New York: Academic Press.

    Google Scholar 

  • Gödel, K. (1930). Die Vollständigkeit der Axiome des logischen Funktionenkalküls. Monatshefte für Mathematik und Physik, 37, 349–360. English translation in Gödel (1986), 102–123.

    Article  MathSciNet  Google Scholar 

  • Gödel, K. (1986). In S. Feferman et al. (Eds.), Collected works, volume I. New York: Oxford University Press.

    Google Scholar 

  • Hanf, W. (1974). Nonrecursive tilings of the plane. I. Journal of Symbolic Logic, 39, 283–285.

    Article  MathSciNet  Google Scholar 

  • Herbrand, J. (1931). Sur le probléme fondamental de la logique mathématique. Sprawozdania z posiedzeǹ Towarzystwa Naukowege Warszawskiego, Wydzial III, 23, 12–56. English translation in Herbrand (1971), 215–271.

    MATH  Google Scholar 

  • Herbrand, J. (1971). In W. D. Goldfarb (Ed.), Logical writings. Dordrecht and Cambridge: Reidel and Harvard University Press.

    Google Scholar 

  • Hilbert, D., & Ackermann, W. (1928). Grundzüge der theoretischen Logik. Berlin: Springer.

    MATH  Google Scholar 

  • Hilbert, D., & Bernays, P. (1939). Grundlagen der Mathematik (Vol. 2). Berlin: Springer.

    MATH  Google Scholar 

  • Jockusch, C. G., Jr., & Soare, R. I. (1972). \(\Pi _{1}^0\) classes and degrees of theories. Transactions of the American Mathematical Society, 173, 33–56.

    MathSciNet  MATH  Google Scholar 

  • Kleene, S. C. (1952). Introduction to metamathematics. Amsterdam: Van Nostrand Reinhold.

    MATH  Google Scholar 

  • Kreisel, G. (1953). Note on arithmetic models for consistent formulae of the preciate calculus II. In Proceedings of the XIth International Congress of Philosophy (pp. 39–49). Amsterdam: North-Holland.

    Google Scholar 

  • Mostowski, A. (1953). On a system of axioms which has no recursively enumerable model. Fundamenta Mathematicae, 40, 56–61.

    Article  MathSciNet  Google Scholar 

  • Mostowski, A. (1955). A formula with no recursively enumerable model. Fundamenta Mathematicae, 43, 125–140.

    Article  MathSciNet  Google Scholar 

  • Putnam, H. (1957). Arithmetic models for consistent formulae of quantification theory. Journal of Symbolic Logic, 22, 110–111.

    Google Scholar 

  • Putnam, H. (1965). Trial and error predicates and the solution to a problem of Mostowski. Journal of Symbolic Logic, 30, 49–57.

    Article  MathSciNet  Google Scholar 

  • Shoenfield, J. R. (1959). On degrees of unsolvability. Annals of Mathematics, 69, 644–653.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Philosophy, Harvard University, Cambridge, MA, 02138, USA

    Warren Goldfarb

Authors
  1. Warren Goldfarb
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Correspondence to Warren Goldfarb .

Editor information

Editors and Affiliations

  1. Department of Philosophy, University of Minnesota Twin Cities, Minneapolis, MN, USA

    Geoffrey Hellman

  2. Department of Philosophy, University of Minnesota Twin Cities, Minneapolis, MN, USA

    Roy T. Cook

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Goldfarb, W. (2018). Putnam’s Theorem on the Complexity of Models. In: Hellman, G., Cook, R. (eds) Hilary Putnam on Logic and Mathematics. Outstanding Contributions to Logic, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-96274-0_4

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