## Gödels reformulation of Gentzen’s first consistency proof for arithmetic: the no-counterexample interpretation (2005)

Venue: | The. Bulletin of Symbolic Logic |

Citations: | 2 - 0 self |

### BibTeX

@ARTICLE{Tait05gödelsreformulation,

author = {W. W. Tait and W. W. Tait},

title = {Gödels reformulation of Gentzen’s first consistency proof for arithmetic: the no-counterexample interpretation},

journal = {The. Bulletin of Symbolic Logic},

year = {2005},

volume = {11}

}

### OpenURL

### Abstract

Abstract. The last section of “Lecture at Zilsel’s ” [9, §4] contains an interesting but quite condensed discussion of Gentzen’s first version of his consistency proof for P A [8], reformulating it as what has come to be called the no-counterexample interpretation. I will describe Gentzen’s result (in game-theoretic terms), fill in the details (with some corrections) of Gödel’s reformulation, and discuss the relation between the two proofs. 1. Let me begin with a description of Gentzen’s consistency proof. As had already been noted in [5], we may express it in terms of a game. 1 To simplify things, we can assume that the logical constants of the classical system of number theory, P A, are ∧, ∨, ∀ and ∃ and that negations are applied only to atomic formulas. ¬φ in general is represented by the complement φ of φ, obtained by interchanging ∧ with ∨, ∀ with ∃, and atomic sentences with their negations. The components of the sentences φ ∨ ψ and φ ∧ ψ are φ and ψ. The components of the sentences ∃xφ(x) and ∀xφ(x) are the sentences φ(¯n) for each numeral ¯n. A ∧- or ∀-sentence, called a �-sentence, is thus expressed by the conjunction of its components and a ∨- or ∃-sentence, called a �-sentence, is expressed by the disjunction of them; and so it follows that every sentence can be represented as an infinitary propositional formula built up from prime sentences— atomic or negated atomic sentences. Disjunctive and conjunctive sentences with the components φn (where the range of n is 1, 2 or ω) will be denoted respectively by

### Citations

72 |
Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics
- Spector
- 1962
(Show Context)
Citation Context ... by h0(m) = 0 for all 8 [12, Remark 3.9] notes that the solution of the corresponding system of equations for φ a Π 0 3 sentence ∀x∃y∀zA(x, y, z) is the same system that is solved by bar recursion in =-=[16]-=- to obtain the Dialectica interpretation of ∀x¬¬∃y∀zA(x, y, z) −→ ¬¬φ. Kohlenbach uses the same construction to obtain the NCI for ψ from witnesses of the NCIs of φ and φ −→ ψ in this case.sm and GÖDE... |

46 |
A semantics of evidence for classical arithmetic
- Coquand
- 1995
(Show Context)
Citation Context ...ails (with some corrections) of Gödel’s reformulation, and discuss the relation between the two proofs. 1. Let me begin with a description of Gentzen’s consistency proof. As had already been noted in =-=[5]-=-, we may express it in terms of a game. 1 To simplify things, we can assume that the logical constants of the classical system of number theory, P A, are ∧, ∨, ∀ and ∃ and that negations are applied o... |

28 |
Zur Widerspruchsfreiheit der Zahlentheorie. Mathematische Annalen 117
- Ackermann
- 1940
(Show Context)
Citation Context ...e could extract a witness. His proof is not based on the game-theoretic idea behind Gentzen’s consistency proof for P A, but rather is a corollary of Ackermann’s proof, using the ɛsubstitution method =-=[1]-=-: Assume given a deduction of φ. From φ logically follows ∃x1 · · · xnA[x1, f ′ 1(x1), . . . , xn, f ′ n(x1, . . . , xn)] In the ɛ-calculus, this has the form A[t1, f ′ 1(t1), . . . , tn, f ′ n(t1, . ... |

23 |
Die Grundlegung der elementaren Zahlenlehre, Mathematische Annalen 104
- Hilbert
- 1931
(Show Context)
Citation Context ..., Gentzen considered a ‘semi-formal’ system which, on the face of it, is stronger than P A: he admitted as axioms finitistically verifiable equations. This seems to be the system Hilbert describes in =-=[10]-=-. There would be no difficulty in extending the present treatment to this ‘system.’ But if we assume that any such equation is derivable in primitive recursive arithmetic, then the greater scope of hi... |

18 | On the no-counterexample interpretation
- Kohlenbach
- 1999
(Show Context)
Citation Context ...ψ, not of φ. Second, the equation is solved (as we shall see) not by recursion on the ordinal α of λg2G1(g), but by recursion in 2 α . In the first published treatment of the NCI for modus ponens, in =-=[12]-=-, the functional N is defined directly by the principle of bar recursion. As noted in [17] and by Kohlenbach, this yields the witness H of ψ as a function of the witnesses F and G of the NCIs of φ and... |

12 |
Beweistheoretische Erfassung der unendlichen Induktion in der Zahlentheorie, Mathematische Annalen 122
- Schütte
- 1951
(Show Context)
Citation Context ... Gentzen assigned an ordinal < ɛ0 to each formal deduction, which turns out to be a bound on the height of the associated tree. The construction of the reduction proceeds by recursion on the ordinal. =-=[15]-=-, at the cost of slightly obscuring (if you don’t read it thoughtfully) the constructive content of Gentzen’s result, recasts the construction as a cut-elimination theorem for P A with the ω-rule, in ... |

9 |
Explaining the Gentzen-Takeuti reduction steps: A second-order system, Archive for Mathematical Logic 40
- Buchholz, Feferman, et al.
- 2001
(Show Context)
Citation Context ... in extending the present treatment to this ‘system.’ But if we assume that any such equation is derivable in primitive recursive arithmetic, then the greater scope of his theorem is only apparent. 4 =-=[4]-=- contains a detailed discussion of the relation between Gentzen’s proof and Schütte’s cut-elimination theorem. 5 For a discussion of the criticism of the original version of Gentzen’s proof, leading t... |

7 |
Die widerspruchfreiheit der reinen zahlentheorie (the consistency of arithmetic). 112(4):493–565
- Gentzen
- 1936
(Show Context)
Citation Context ...NTERPRETATION W. W. TAIT Abstract. The last section of “Lecture at Zilsel’s” [9, §4] contains an interesting but quite condensed discussion of Gentzen’s first version of his consistency proof for P A =-=[8]-=-, reformulating it as what has come to be called the no-counterexample interpretation. I will describe Gentzen’s result (in game-theoretic terms), fill in the details (with some corrections) of Gödel’... |

5 |
Lecture at Zilsel’s
- Gödel
- 1995
(Show Context)
Citation Context ...did not assign ordinals to deductions: in place of recursion on ordinals, it uses recursion on well-founded trees (essentially the principle of bar recursion). 5 4. Gödel describes Gentzen’s proof in =-=[9]-=- in a way that avoids a calculus of sequents (which, as I mentioned, we have replaced by finite sets). He considers deductions of single formulas in an ordinary Hilbert-style formalization of P A. Ass... |

4 | Update procedures and the 1-consistency of arithmetic
- Avigad
- 2002
(Show Context)
Citation Context ... and therefore not expect the same level of care as in a published paper. However, another possibility in the case of the first inaccuracy is that he confused two notions of recursive definition of 9 =-=[2]-=- solves the equation L(hn) = L(hn+1) in essentially the same way as [18], as an instance of constructing a finite fixed point of an update procedure. The update procedure in this case is F (h) = 〈L(h)... |

4 |
On the interpretation of non-finitistproofs’,partI,The
- Kreisel
- 1951
(Show Context)
Citation Context ...ing strategy F for � (in the restricted game) is precisely a witness for the so-called no-counterexample interpretation, NCI ∃F ∀fA[Fi(f), fj(F1(f), . . . , Fj(f))] of φ, first introduced in print in =-=[13]-=-. Kreisel showed that, from any deduction of φ, one could extract a witness. His proof is not based on the game-theoretic idea behind Gentzen’s consistency proof for P A, but rather is a corollary of ... |

3 |
On the Original Gentzen Consistency Proof for Number Theory, in Intuitionism and Proof Theory, Kino, Myhill & Vesley eds, pp 409-417. Original proof printed in the The Collected Papers of Gerhard Gentzen by
- Bernays
- 1969
(Show Context)
Citation Context ...e cut-free deduction obtained from the given deduction is the tree determined by Gentzen’s winning strategy. 4 An original version of Gentzen’s proof, which he did not publish but is now published in =-=[3]-=-, did not assign ordinals to deductions: in place of recursion on ordinals, it uses recursion on well-founded trees (essentially the principle of bar recursion). 5 4. Gödel describes Gentzen’s proof i... |

1 | editor), Recursive function theory, proceedings of symposia in pure mathematics - Dekker - 1962 |

1 | Gentzens problem: Mathematische logik im nationalsozialistischen deutschland - Menzler-Trott - 2001 |

1 |
The no counterexample interpretation for arithmetic, Unpublished manuscript
- Tait
(Show Context)
Citation Context ...of P A extended by numerical function variables. To see the necessity of definition by cases, consider a 6 A direct proof of the NCI for P A based on that lemma is given in the unpublished manuscript =-=[17]-=- dating from the early 1960’s and listed among the references in [19] as “To Appear.”sGÖDEL’S REFORMULATION OF GENTZEN’S FIRST CONSISTENCY PROOF 7 deduction of ∀xA(x) ∨ ∀yB(y) −→ ∀zC(z) from ∀xA(x) −→... |

1 |
defined by transfinite recursion, The
- Functionals
- 1965
(Show Context)
Citation Context ...btained from a deduction of Γ by recursion on the reductions (as well-founded trees) of the premises from which Γ is obtained. The method of constructing H in [17], drawing on the machinery set up in =-=[18]-=-, is not uniform in F and G; but it does yield the ordinal of the functionals in H from those of the functionals in F and G, and so more parallels the published version of Gentzen’s proof. Moreover, i... |

1 |
unpublished papers on foundations of mathematics
- Gödel’s
(Show Context)
Citation Context ... denoted respectively by � � n φn I am grateful to Jeremy Avigad, John W. Dawson Jr., and Solomon Feferman for valuable comments on earlier drafts of this paper. 1 This paper is an expansion of §4 of =-=[20]-=-. I regret that I did not know Thierry Coquand’s paper when I wrote [20] and so failed to cite it. 1 n φns2 W. W. TAIT 2. There is of course one well-known game T (φ) associated with sentences φ of P ... |