## A BUCHOLZ DERIVATION SYSTEM FOR THE ORDINAL ANALYSIS OF KP + Π3-REFLECTION

### BibTeX

@MISC{Michelbrink_abucholz,

author = {Markus Michelbrink},

title = {A BUCHOLZ DERIVATION SYSTEM FOR THE ORDINAL ANALYSIS OF KP + Π3-REFLECTION},

year = {}

}

### OpenURL

### Abstract

Abstract. In this paper we introduce a notation system for the infinitary derivations occurring in the ordinal analysis of KP + Π3-Reflection due to Michael Rathjen. This allows a finitary ordinal analysis of KP+Π3-Reflection. The method used is an extension of techniques developed by Wilfried Buchholz, namely operator controlled notation systems for RS ∞-derivations. Similarly to Buchholz we obtain a characterisation of the provably recursive functions of KP+Π3-Reflection as <-recursive functions where < is the ordering on Rathjen’s ordinal notation system T (K). Further we show a conservation result for Π 0 2-sentences. §1. Introduction. Ordinal analysis uses cut-elimination techniques for proof theoretic investigations. The termination of the cut-elimination process is guaranteed by assigning decreasing ordinals to the proofs emerging in the process. Gerhard Gentzen was the first to form a relationship between an ordinal ε0 and a foundational mathematical theory (nowadays denoted Peano Arithmetic PA)

### Citations

837 |
Theory of Recursive Functions and Effective Computability
- Rogers
- 1967
(Show Context)
Citation Context ...finite sets are exactly the Σ1-definable subsets of the natural numbers (Barwise [3]) and a partial function is recursive if and only if the graph of the function is recursive enumerable (e.g. Rogers =-=[15]-=-) we may interprete this as a characterisation of the provably recursive (provably total) functions of KP + Π3-Refl. In this section only we use transfinite induction. Since we proceed again in a way ... |

487 |
Set theory
- Jech
- 1997
(Show Context)
Citation Context ...together with a well ordering < on T (K) which is primitive recursive as well. We assume in this section the existence of a weakly compact cardinal K. Weakly compact cardinals are Π 1 1-indescribable =-=[14]-=-. This property of K is used only once. All following theorems are quoted from [20]. We are only interested in the properties stated. Therefore we refrain from giving the exact definitions and proofs.... |

191 |
Admissible Sets and Structures
- Barwise
- 1975
(Show Context)
Citation Context ...fined by <-recursion in the sense of Takeuti [28], the recursive enumerable subsets in the structure of the hereditary finite sets are exactly the Σ1-definable subsets of the natural numbers (Barwise =-=[3]-=-) and a partial function is recursive if and only if the graph of the function is recursive enumerable (e.g. Rogers [15]) we may interprete this as a characterisation of the provably recursive (provab... |

107 |
Metamathematics of first-order arithmetic
- Hájek, Pudlák
- 1993
(Show Context)
Citation Context ...ead of “the” here and at some other places. From an extensional point of view there is only one function. 3 Such formulations which I think are rather ill-chosen are common in the literature see e.g. =-=[11, 29]-=-. However they allow to emphasise the double rôle of x (as metavariable and variable of the term) by the formulation A( ˙x).44 MARKUS MICHELBRINK Therefore we may argue in PRA as follows: 1 = true0(φ... |

45 |
Proof Theory. An Introduction
- Pohlers
- 1989
(Show Context)
Citation Context ...proofs. The reader may find them in [10, 16]. The properties stated here are used heavily in the proof of Theorem 12.4. We start by fixing some basic notions. Our main references for this section are =-=[26, 18, 20]-=-. We use the notations On, Card, Lim for the classes of ordinals, infinite cardinals and limit ordinals respectively. Small Greek letters are reserved for ordinals with one exception: ϕ which is used ... |

33 | Schütte: Proof theory of impredicative subsystems of analysis. (Bibliopolis - Buchholz, K - 1988 |

33 |
Set Theory: An introduction to large cardinals
- Drake
- 1974
(Show Context)
Citation Context ...only once. All following theorems are quoted from [20]. We are only interested in the properties stated. Therefore we refrain from giving the exact definitions and proofs. The reader may find them in =-=[10, 16]-=-. The properties stated here are used heavily in the proof of Theorem 12.4. We start by fixing some basic notions. Our main references for this section are [26, 18, 20]. We use the notations On, Card,... |

32 |
Eine beweistheoretische Untersuchung von (Δ 1 2-CA) + (BI) und verwandter Systeme, Bayerische Akademie der Wissenschaften, Sitzungsberichte
- Jäger, Pohlers
- 1983
(Show Context)
Citation Context ...e length of the derivation. This work clarified the relationship between ε0 and PA. Since this time infinitary methods have been successfully applied for the analysis of numerous other theories (e.g. =-=[13, 9, 20, 22, 23]-=- to name just a few). However as pointed out by Wilfried Buchholz [5] something is lost by passing from finite to infinite derivations. So Gentzen’s method gives us bounds for the provably recursive f... |

15 |
Begrundung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Verenderlichen mit unendlichem Ausdehnungsbereich, Videnskapsselskapets skrifter
- Skolem
- 1923
(Show Context)
Citation Context ...n symbol and the symbol 0 the only constant symbol of the language. In PRA natural induction is restricted to formulas without quantifiers i.e. ∆0-formulas of the language. Skolem introduced PRA 1923 =-=[27]-=- as an informal system (without quantifiers). The theory is discussed in length in Hilbert/Bernays [12] and serves as an example for finitary reasoning. The quantifier-free part of the theory has a lo... |

12 |
Proof theory for theories of ordinals I: recursively Mahlo ordinals, Annals of Pure and applied Logic 122
- Arai
- 2003
(Show Context)
Citation Context ...he infinitary ordinal analysis of Π 1 2-CA due to Michael Rathjen. There is as well the hope that this work may contribute in comparing the work of Michael Rathjen [22, 23, 24, 19] and Toshiyasu Arai =-=[1, 2]-=-. The paper is organised as follows: The first three sections are completely devoted to the citation of definitions and results. In section 2 we recall definitions and properties of Rathjens ordinal n... |

12 |
Beweistheoretische Erfassung der unendlichen Induktion in der Zahlentheorie, Mathematische Annalen 122
- Schütte
- 1951
(Show Context)
Citation Context ...ging in the process. Gerhard Gentzen was the first to form a relationship between an ordinal ε0 and a foundational mathematical theory (nowadays denoted Peano Arithmetic PA) in this way. Kurt Schütte =-=[25]-=- showed that cut-elimination can be radically simplified by moving to an infinitary proof calculus which allows the embedding of PA. This is made possible by replacing the generalisation rule by the i... |

7 |
Troelstra: Metamathematical investigations of intuitionistic arithmetic and analysis
- S
- 1973
(Show Context)
Citation Context ...ead of “the” here and at some other places. From an extensional point of view there is only one function. 3 Such formulations which I think are rather ill-chosen are common in the literature see e.g. =-=[11, 29]-=-. However they allow to emphasise the double rôle of x (as metavariable and variable of the term) by the formulation A( ˙x).44 MARKUS MICHELBRINK Therefore we may argue in PRA as follows: 1 = true0(φ... |

4 |
The sizes of the indescribable cardinals
- Levy
- 1971
(Show Context)
Citation Context ...only once. All following theorems are quoted from [20]. We are only interested in the properties stated. Therefore we refrain from giving the exact definitions and proofs. The reader may find them in =-=[10, 16]-=-. The properties stated here are used heavily in the proof of Theorem 12.4. We start by fixing some basic notions. Our main references for this section are [26, 18, 20]. We use the notations On, Card,... |

4 |
An ordinal analysis of parameter free Π 1 2 - comprehension
- Rathjen
(Show Context)
Citation Context ...tic way to a finitary treatment of the infinitary ordinal analysis of Π 1 2-CA due to Michael Rathjen. There is as well the hope that this work may contribute in comparing the work of Michael Rathjen =-=[22, 23, 24, 19]-=- and Toshiyasu Arai [1, 2]. The paper is organised as follows: The first three sections are completely devoted to the citation of definitions and results. In section 2 we recall definitions and proper... |

3 |
A simplified version of local predicativity, Proof Theory
- Buchholz
- 1990
(Show Context)
Citation Context ...ahlo numbers are stationary in τ. Further we introduced the conditions h),i),k) and the requirement o(d[i]) ∈ H(k(d) ∪ k(i)). The concept of operator controlled derivations is as well due to Buchholz =-=[4]-=-. If (D, o, deg, ref, tp, []) is a notation system for RS(K)-derivations then we can unfold d ∈ D by transfinite recursion over o(d) to an infinitary RS(K)-derivation d ∞ := tp(d)(d[i] ∞ ) i∈|tp(d)|. ... |

3 |
theory of reflection, Annals of Pure and
- Proof
- 1994
(Show Context)
Citation Context ...e length of the derivation. This work clarified the relationship between ε0 and PA. Since this time infinitary methods have been successfully applied for the analysis of numerous other theories (e.g. =-=[13, 9, 20, 22, 23]-=- to name just a few). However as pointed out by Wilfried Buchholz [5] something is lost by passing from finite to infinite derivations. So Gentzen’s method gives us bounds for the provably recursive f... |

2 |
Zur endlichen Behandlung der Beweistheorie schwacher Fragmente der Mengenlehre
- MICHELBRINK
- 2000
(Show Context)
Citation Context ...5, 03D20. Key words and phrases. Finitary Proof Theory, Ordinal Analysis, Impredicative Theories. Supported by EPSRC grant GR/S30450/01. The results in this paper are part of the authors Ph.D. thesis =-=[17]-=-. 12 MARKUS MICHELBRINK the other side is not completely trivial, and it seems to me that all presentations of this subject in the existing literature are more ore less unsatisfactory”. We agree to t... |

1 | theory for theories of ordinals II: Π-3-reflection - Proof |

1 |
advances in ordinal analysis:Π1 2 − CA and related systems
- Recent
- 1995
(Show Context)
Citation Context ...e length of the derivation. This work clarified the relationship between ε0 and PA. Since this time infinitary methods have been successfully applied for the analysis of numerous other theories (e.g. =-=[13, 9, 20, 22, 23]-=- to name just a few). However as pointed out by Wilfried Buchholz [5] something is lost by passing from finite to infinite derivations. So Gentzen’s method gives us bounds for the provably recursive f... |

1 |
Finitary reductions for local predicativity, 1: recursively regular ordinals, Logic Colloquium 98
- Tupailo
(Show Context)
Citation Context ...hat all presentations of this subject in the existing literature are more ore less unsatisfactory”. We agree to the latter as well. In this paper we continue work of Buchholz [5, 6, 7, 8] and Tupailo =-=[32]-=-. We define a finitary notation system for the infinitary derivations occurring in the ordinal analysis of KP + Π3-Reflection due to Michael Rathjen [20, 22]. This gives a finitary ordinal analysis of... |