Abstract
Hilbert's ε-calculus is based on an extension of the language of predicate logic by a term-forming operator e x . Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand's Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure.
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References
Ackermann, W., ‘Begrundung des Tertium non datur mittels der Hilbertschen The-orie der Widerspruchsfreiheit’, Mathematische Annalen 93: 1–36, 1925.
Ackermann, W., ‘Zur Widerspruchsfreiheit der Zahlentheorie’, Mathematische Annalen 117: 162–194, 1940.
Arai, T., ‘Epsilon substitution method for \(ID_1(\prod^{0}_{1} \vee \sum^{0}_{1})\) ’, Annals of Pure and Applied Logic 121: 163–208, 2003.
Arai, T., ‘Ideas in the epsilon substitution method for \(\prod^{0}_{1}\)-FIX’, Annals of Pure and Applied Logic 136: 3–21, 2005.
Avigad, J., ‘Update procedures and the 1-consistency of arithmetic’, Mathematical Logic Quarterly 48: 3–13, 2002.
Baaz, M., and A. Leitsch, ‘On Skolemization and proof complexity’, Fundamenta Informaticae 20: 353–379, 1994.
Bell, J. L., ‘Hubert's epsilon-operator and classical logic’, Journal of Philosophical Logic 22: 1–18, 1993.
Bell, J. L., ‘Hilbert's epsilon operator in intuitionistic type theories’, Mathematical Logic Quarterly 39: 323–337, 1993.
Blass, A., and Y. Gurevich, ‘The logic of choice’, Journal of Symbolic Logic 65: 1264–1310, 2000.
Buss, S. R., ‘Introduction to proof theory’, in S. R. Buss, (ed.), Handbook of Proof Theory, Elsevier, 1998, pp. 1–79.
DeViDi, D., ‘Intuitionistic epsilon- and tau-calculi’, Mathematical Logic Quarterly 41: 523–546, 1995.
Ebbinghaus, H.-D., J. Flum, and W. Thomas, Mathematical Logic, Springer, 1994.
Fitting, M., ‘A modal logic epsilon-calculus’, Notre Dame Journal of Formal Logic 16: 1–16, 1975.
Hilbert, D., and P. Bernays, Grundlagen der Mathematik, vol. 2, Springer, Berlin, 1939.
Kreisel, G., ‘Interpretation of non-flnitist proofs II, Journal of Symbolic Logic 17: 43–58, 1952.
Leisenring, A., Mathematical Logic and Hilbert's ε-symbol, MacDonald Technical and Scientific, London, 1969.
Mints, G., ‘A termination proof for epsilon substitution using partial derivations’, Theoretical Computer Science 303: 187–213, 2003.
Mints, G., and S. Tupailo, ‘Epsilon-substitution method for the ramified language and Δ11-comprehension rule’, in A. Cantini et al., (eds.), Logic and Foundations of Mathematics, Kluwer, Dordrecht, 1999, pp. 107–130.
Moser, G., ‘Ackermann's substitution method (remixed)’, Annals of Pure and Applied Logic 2006, to appear.
MOSTOWSKI, A., ‘The Hilbert epsilon function in many-valued logics’, Acta Philos. Fenn. 16: 169–188, 1963.
Orevkov, V. P., ‘Lower bounds for increasing complexity of derivations after cut elimination’, Journal of Soviet Mathematics 20: 2337–2350, 1982.
Statman, R., ‘Lower bounds on Herbrand's theorem’, Proceedings of the American Mathematical Society 75: 104–107, 1979.
Troelstra, A., and H. Schwichtenberg, Basic Proof Theory, Cambridge University Press, 2nd ed., 2000.
Von Neumann, J., ‘Zur Hilbertschen Beweistheorie’, Mathematische Zeitschrift 26: 1–46, 1927.
Zach, R., ‘The practice of flnitism. Epsilon calculus and consistency proofs in Hilbert's program’, Synthese 137: 211–259, 2003.
Zach, R., ‘Hilbert's “Verungliickter Beweis”, the first epsilon theorem, and consisteny proofs’, History and Philosophy of Logic 25: 79–94, 2004.
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Moser, G., Zach, R. The Epsilon Calculus and Herbrand Complexity. Stud Logica 82, 133–155 (2006). https://doi.org/10.1007/s11225-006-6610-7
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DOI: https://doi.org/10.1007/s11225-006-6610-7