The Epsilon Calculus and Herbrand Complexity

Hilbert's ε-calculus is based on an extension of the language of predicate logic by a term-forming operator ex. 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.

[1]  Georg Moser,et al.  Ackermann's substitution method (remixed) , 2006, Ann. Pure Appl. Log..

[2]  L. M. Milne-Thomson,et al.  Grundlagen der Mathematik , 1935, Nature.

[3]  Wilhelm Ackermann,et al.  Begründung des „tertium non datur” mittels der Hilbertschen Theorie der Widerspruchsfreiheit , 1925 .

[4]  Grigori Mints,et al.  A termination proof for epsilon substitution using partial derivations , 2003, Theor. Comput. Sci..

[5]  John L. Bell,et al.  Hilbert's ɛ-operator and classical logic , 1993, J. Philos. Log..

[6]  Lev Gordeev,et al.  Basic proof theory , 1998 .

[7]  G. A. Miller,et al.  MATHEMATISCHE ZEITSCHRIFT. , 1920, Science.

[8]  Andreas Blass,et al.  The Logic of Choice , 2000, Journal of Symbolic Logic.

[9]  Toshiyasu Arai,et al.  Ideas in the epsilon substitution method for II 1 0-FIX , 2005, Ann. Pure Appl. Log..

[10]  S. Shelah,et al.  Annals of Pure and Applied Logic , 1991 .

[11]  Toshiyasu Arai,et al.  Epsilon substitution method for ID1(Pi10 or Sigma10) , 2003, Ann. Pure Appl. Log..

[12]  Alexander Leitsch,et al.  On Skolemization and Proof Complexity , 1994, Fundam. Informaticae.

[13]  Samuel R. Buss,et al.  Chapter I - An Introduction to Proof Theory , 1998 .

[14]  Georg Kreisel,et al.  On the interpretation of non-finitist proofs—Part I , 1951, Journal of Symbolic Logic.

[15]  Grigori Mints,et al.  Epsilon-Substitution Method for the Ramified Language and $$\Delta _1^1$$-Comprehension Rule , 1999 .

[16]  Andrzej Mostowski The Hilbert Epsilon Function in Many-Valued Logics , 1979 .

[17]  John L. Bell,et al.  Hilbert's ϵ-Operator in Intuitionistic Type Theories , 1993, Math. Log. Q..

[18]  David DeVidi Intuitionistic epsilon- and tau-calculi , 1995, Math. Log. Q..

[19]  Richard Zach Hilbert's ‘Verunglückter Beweis’, the first epsilon theorem, and consistency proofs , 2002 .

[20]  Melvin Fitting,et al.  A modal logic ε-calculus , 1975, Notre Dame J. Formal Log..

[21]  Grigori Mints,et al.  Epsilon – Substitution Method for the Ramified Language and ∆ 11-Comprehension Rule , 1998 .

[22]  David DeVidi,et al.  Intuitionistic ϵ‐ and τ‐calculi , 1995 .

[23]  Richard Zach The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program , 2004, Synthese.

[24]  J. Neumann Zur Hilbertschen Beweistheorie , 1927 .

[25]  Wilhelm Ackermann,et al.  Zur Widerspruchsfreiheit der Zahlentheorie , 1940 .

[26]  T. Arai Epsilon substitution method for ID_1 (II^0_1 ∨ Σ^0_1) , 2003 .

[27]  R. Statman Lower bounds on Herbrand’s theorem , 1979 .

[28]  J. Avigad Update Procedures and the 1-Consistency of Arithmetic , 2002, Math. Log. Q..

[29]  V. Orevkov Lower bounds for increasing complexity of derivations after cut elimination , 1982 .

[30]  A. Leisenring Mathematical logic and Hilbert's ε-symbol , 1971 .