## 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)

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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 ... |

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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... |

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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... |

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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... |

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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 |
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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... |