Journal of Mathematical Sciences

, Volume 87, Issue 1, pp 3234–3252 | Cite as

A method of epsilon substitution for the predicate logic with equality

  • G. E. Mints


The method of epsilon substitution was defined for arithmetic with interpretation of αxA(x) as the least x satisfying A(x). It proceeds by a series of finite approximations “from below” to a solution of a fixed system of critical formulas. For the predicate logic only approach “from above” similar to cut-elimination was available. We present a definition of epsilon substitution for the predicate logic, prove the termination of the substitution process, and derive the corresponding Herbrand-type theorem. Bibliography: 18 titles.


Predicate Logic Fixed System Substitution Process Finite Approximation Critical Formula 


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  1. 1.
    W. Ackermann, “Begründung des Tertium non datur mittels der Hilbertschen Theorie der Widerspruchsfreiheit,”Math. Ann.,93, 1–36 (1925).CrossRefMathSciNetGoogle Scholar
  2. 2.
    W. Ackermann, “Zur Widerspruchsfreiheit der Zahlentheorie,”Math. Ann.,117, 162–194 (1940).CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    N. Bourbaki,Theorie des Ensembles, Hermann (1958).Google Scholar
  4. 4.
    G. Gentzen, “Die Widerspruchsfreiheit der reinen Zahlentheorie,”Math. Ann.,112, 493–565 (1936).CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    D. Hilbert, “Probleme der Grundlegung der Mathematik,”Math. Ann.,102, 1–9 (1929).MATHMathSciNetGoogle Scholar
  6. 6.
    D. Hilbert and P. Bernays,Grundlagen der Mathematik, Bd. 2, Springer (1970).Google Scholar
  7. 7.
    G. Kreisel, “On the interpretation of non-finitist proofs I,”J. Symbolic Logic,16, 241–267 (1951).MATHMathSciNetGoogle Scholar
  8. 8.
    G. Kreisel, “On the interpretation of non-finitist proofs II,”J. Symbolic Logic,17, 43–58 (1952).MATHMathSciNetGoogle Scholar
  9. 9.
    G. Mints, “Simplified consistency proof for arithmetic,”Proc. Eston. Acad. Sci. Phys. Math.,31, 376–382 (1984).MathSciNetGoogle Scholar
  10. 10.
    G. Mints, “Epsilon substitution method for the theory of hereditarily finite sets,”Proc. Eston. Acad. Sci. Phys. Math.,38, 154–164 (1989).MATHMathSciNetGoogle Scholar
  11. 11.
    G. Mints, “Gentzen-type systems and Hilbert's epsilon substitution method. I,” in:Logic, Method. and Philos. of Sci, IX, Elsevier (1994), pp. 91–122.Google Scholar
  12. 12.
    G. Mints, S. Tupailo, and W. Buchholz, “Epsilon substitution method for elementary analysis”, to appear inArchive Math. Logic. Google Scholar
  13. 13.
    G. Mints, “Strong termination proof for the epsilon substitution method,” to appear inJ. Symbolic Logic.Google Scholar
  14. 14.
    J. von Neumann, “Zur Hilbertshen Beweistheorie,”Math. Z.,26, 1–46 (1927).CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    M. Rogava, “On sequential variants of applied predicate calculi,”Zap Nauchn. Semin. LOMI, 4 (1967).Google Scholar
  16. 16.
    K. Schütte,Proof Theory, Springer (1977).Google Scholar
  17. 17.
    W. Tait, “The substitution method,”J. Symbolic Logic,30, 175–192 (1965).MATHMathSciNetGoogle Scholar
  18. 18.
    H. Weyl, “David Hilbert and his mathematical work,”Bull. Am. Math. Soc.,50, 612–654 (1944).MATHMathSciNetGoogle Scholar

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© Plenum Publishing Corporation 1997

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  • G. E. Mints

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