## Dedicated to Wolfram Pohlers on his retirement

### BibTeX

@MISC{Buchholz_dedicatedto,

author = {Wilfried Buchholz},

title = {Dedicated to Wolfram Pohlers on his retirement},

year = {}

}

### OpenURL

### Abstract

One of the major problems in reductive proof theory in the early 1970s was to give a proof-theoretic reduction of classical theories of iterated arithmetical inductive definitions to corresponding constructive systems. This problem was solved in [BFPS] in various ways which all where based on the method of cut-elimination (normalization, reps.) for infinitary Tait-style sequent calculi (infinitary systems of natural deduction,

### Citations

70 |
Iterated Inductive Definitions and Subsystems of Analysis
- Buchholz
- 1981
(Show Context)
Citation Context ...heory in the early 1970s was to give a proof-theoretic reduction of classical theories of iterated arithmetical inductive definitions to corresponding constructive systems. This problem was solved in =-=[BFPS]-=- in various ways which all where based on the method of cut-elimination (normalization, reps.) for infinitary Tait-style sequent calculi (infinitary systems of natural deduction, resp.). Only quite re... |

59 |
Die Widerspruchsfreiheit der reinen Zahlentheorie
- Gentzen
(Show Context)
Citation Context ... reasons which, as we hope, justify a publication of this additional proof. First, it is considerably more direct then all the existing ones. Second, the method used here stems to a great extent from =-=[Ge36]-=- and therefore may be interesting for historical reasons too. Actually I have used a variant of this method under the label “notations for infinitary derivations” already in several papers (e.g. [Bu91... |

27 |
Notation systems for infinitary derivations
- Buchholz
- 1991
(Show Context)
Citation Context ...Ge36] and therefore may be interesting for historical reasons too. Actually I have already used a variant of this method under the label “notations for infinitary derivations” in several papers (e.g. =-=[Bu91]-=-, [Bu97], [Bu01]) without mentioning its close relationship to [Ge36]. When writing [Bu91] I was definitely not aware of this connection; but cf. [Bu95]. The method from [Ge36] can be roughly describe... |

20 |
Neue Fassung des Widerspruchsfreiheitsbeweises fur die reine Zahlentheorie. Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften. Neue Folge 4
- Gentzen
- 1938
(Show Context)
Citation Context ...ch is available in ID i ν+1(W) but not in ID i ν(W). We overcome this difficulty by using (a generalization of) Gentzen’s technique for proving transfinite induction up to ordinals < ε0 within Z (cf. =-=[Ge43]-=-). In order to avoid some annoying but inessential technical complications we restrict our treatment to ν < ω. So in the whole paper ν is a fixed natural number > 0. 1Preliminaries. For the reader’s ... |

16 | Explaining Gentzen's Consistency Proof within in Proof Theory
- Buchholz
- 1997
(Show Context)
Citation Context ...d therefore may be interesting for historical reasons too. Actually I have already used a variant of this method under the label “notations for infinitary derivations” in several papers (e.g. [Bu91], =-=[Bu97]-=-, [Bu01]) without mentioning its close relationship to [Ge36]. When writing [Bu91] I was definitely not aware of this connection; but cf. [Bu95]. The method from [Ge36] can be roughly described as fol... |

11 | Explaining the Gentzen-Takeuti reduction steps
- Buchholz
- 1997
(Show Context)
Citation Context ...ore may be interesting for historical reasons too. Actually I have already used a variant of this method under the label “notations for infinitary derivations” in several papers (e.g. [Bu91], [Bu97], =-=[Bu01]-=-) without mentioning its close relationship to [Ge36]. When writing [Bu91] I was definitely not aware of this connection; but cf. [Bu95]. The method from [Ge36] can be roughly described as follows: By... |

8 | Functional interpretation and inductive definitions
- Avigad, Towsner
(Show Context)
Citation Context ... where based on the method of cut-elimination (normalization, reps.) for infinitary Tait-style sequent calculi (infinitary systems of natural deduction, resp.). Only quite recently Avigad and Towsner =-=[AT09]-=- succeeded in giving a reduction of classical iterated ID theories to constructive ones by the method of functional interpretation. For a thorough exposition and discussion of all this cf. [Fef]. In t... |

1 |
The Ωµ+1-rule, in Buchholz et al
- Buchholz
- 1981
(Show Context)
Citation Context ...)-inference (Lemma 1). On first Γ(h) sight the present system ID ∞ ν looks exactly like the system ID ∞ ν in [Bu02] (which itself is the Tait-style version of the natural deduction system ID ∞ ν from =-=[Bu81]-=-), but there is some subtle difference concerning the index sets | ˜ ΩP | of instances of the Ω-rule. In [Bu02], | ˜ ΩP | is a set of infinitary derivations while in the present paper | ˜ ΩP | is a se... |

1 |
On Gentzen’s consistency proofs for arithmetic. Oberwolfach
- Buchholz
- 1995
(Show Context)
Citation Context ...infinitary derivations” in several papers (e.g. [Bu91], [Bu97], [Bu01]) without mentioning its close relationship to [Ge36]. When writing [Bu91] I was definitely not aware of this connection; but cf. =-=[Bu95]-=-. The method from [Ge36] can be roughly described as follows: By (primitive) recursion on the build-up of h, for each derivation h in a suitably designed finitary proof system Z of first order arithme... |

1 | Assigning ordinals to proofs in a perspicious way
- Buchholz
(Show Context)
Citation Context ...ion over the relation ≺ := {(h[i], h) : h ∈ Z & i ∈ Ih}. In the present paper we proceed similarly. Let IDν be the finitary Tait-style system of ν-fold iterated inductive definitions as introduced in =-=[Bu02]-=-. We extend IDν by certain inferences E, Dσ, S Π P,F (which do not alter the set of derivable sequents) to a finitary system ID ∗ ν. This step corresponds very much to the passage from BI − 1 to BI∗1 ... |