## Proof-theoretic investigations on Kruskal's theorem (1993)

Venue: | Ann. Pure Appl. Logic |

Citations: | 23 - 3 self |

### BibTeX

@ARTICLE{Rathjen93proof-theoreticinvestigations,

author = {Michael Rathjen and Andreas Weiermann},

title = {Proof-theoretic investigations on Kruskal's theorem},

journal = {Ann. Pure Appl. Logic},

year = {1993},

volume = {60},

pages = {49--88}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we calibrate the exact proof--theoretic strength of Kruskal's theorem, thereby giving, in some sense, the most elementary proof of Kruskal's theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction. Introduction S.G. Simpson in his article [10], "Nonprovability of certain combinatorial properties of finite trees", presents proof--theoretic results, due to H. Friedman, about embeddability properties of finite trees. It is shown there that Kruskal's theorem is not provable in ATR 0 . An exact description of the proof--theoretic strength of Kruskal's theorem is not given. On the assumption that there is a bad infinite sequence of trees, the usual proof of Kruskal's theorem utilizes the existence of a minimal bad sequence of trees, thereby employing some form of \Pi 1 1 comprehension. So the question arises whether a more constructive proof can be given. The need for a more elementary proof of Kruskal's theorem is especially felt ...

### Citations

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(Show Context)
Citation Context ... of (\Pi 1 2 \Gamma BI) 0 are defined inductively, as usual. D; D 0 ; D 0 ; : : : range as syntactic variables over (\Pi 1 2 \Gamma BI) 0 derivations. All this is completely standard, and we refer to =-=[9] for notions like "length-=- of a derivation D" (abbreviated by j D j), "last inference of D", "direct subderivation of D ". We write D ` \Gamma to mean that D is a derivation of \Gamma. The most importa... |

58 |
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(Show Context)
Citation Context ... Pr (\Pi 1 2 \GammaBI) 0 denotes the \Sigma 0 1 provability predicate for (\Pi 1 2 \Gamma BI) 0 and pF (n)q signifies the Godel number of F (a) when a is replaced by the n th numeral (for details see =-=[11]-=-). The proof of Theorem 10.1 can be formalized in ACA 0 . Therefore ACA 0 ` 8xPr (\Pi 1 2 \GammaBI) 0 (WF (#\Omega x )). So we come to see the following. Theorem 11.1 ACA 0 + RFN \Pi 1 1 ((\Pi 1 2 \Ga... |

46 |
Proof Theory - An Introduction
- Pohlers
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(Show Context)
Citation Context ...l 2 C n (ff) =) ! fi + fl 2 C n+1 (ff); 3. fi 2 C(fi) " ff " C n (ff) =) /fi 2 C n+1 (ff); 4. C(ff) := S fC n (ff) : n ! !g: /ff := minf :s62 C(ff)g: The following Lemmata can be gathered fr=-=om [1] or [4]. Lemma 3.-=-1 /ff !\Omega : Lemma 3.2 1. /ff = C(ff) "\Omega ; 2. ff 2 C(ff) " fi =) /ff ! /fi; 3. ff; fi ! /fl =) ! ff + fi ! /fl: Definition 3.2 Inductive definition of a set T (/) of ordinals and a n... |

37 |
Nonprovability of certain combinatorial properties of finite trees
- Simpson
- 1985
(Show Context)
Citation Context ...g, in some sense, the most elementary proof of Kruskal's theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction. Introduction S.G. Simpson in his article =-=[10], "No-=-nprovability of certain combinatorial properties of finite trees", presents proof--theoretic results, due to H. Friedman, about embeddability properties of finite trees. It is shown there that Kr... |

33 |
Schütte: Proof theory of impredicative subsystems of analysis. (Bibliopolis
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- 1988
(Show Context)
Citation Context ...erspicious proof-theoretic analysis of 17 \Pi 1 2 bar induction we need another concept for representing the HowardBachmann -ordinal namely the /--function which is due to Buchholz. See, for example, =-=[1]-=- for a definition. Let \Omega\Gammat ; !) :=\Omega ! and\Omega\Gamma n + 1; !) :=\Omega \Omega\Gamma n;!) . We are going to show that #(\Omega\Gamma n; !)) = /(\Omega\Gamma n + 1; !)) is true for ever... |

14 |
Well-partial Orderings and Their Maximal Order Types
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(Show Context)
Citation Context ...ga ! \Delta ff) 2 T . This will imply Kruskal's theorem by taking the domain of Xto be a singleton, i.e., ff = 1. For technical reasons we introduce the following terminology, which is due to Schmidt =-=[6]-=-. Definition 2.1 Let X i = hX i ;si i (i = 0; : : : ; n) be pairwise disjoint quasi-- orders and let ff 0 ; : : : ; ff n ordinals such that 0 ! ff 0 ! : : : ! ff ns!. Let X:= X 0 \Phi : : : \Phi X n .... |

13 |
Eine in der reinen Zahlentheorie unbeweisbarer Satz über endliche Folgen von natürlichen Zahlen
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- 1985
(Show Context)
Citation Context ...-sequence 1si 1 ! : : : ! i msn such that x lsx 0 i l for every 1slsm: 1 If X 0 and X 1 are well--quasi--orders then X 0 \Phi X 1 ; X 0\Omega X 1 and X !! 0 are well--quasi--orders, too. According to =-=[8]-=-, this can be shown in ACA 0 (especially, Higman's lemma is provable in ACA 0 ). A finite tree T with labels in a quasi-order X= hX; i is an ordered tripel hT ;sT ; l T i such that hT ;sT i is a finit... |

12 | Proof-theoretic techniques for term rewriting theory - Dershowitz, Okada - 1988 |

4 | Fragments of Kripke-Platek set theory with infinity
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- 1992
(Show Context)
Citation Context ... +\Pi n \GammaF oundation j= #\Omega\Gamma n\Gamma1; !): 2. j (\Pi 1 n \Gamma BI) j=j (\Pi 1 n \Gamma BI) \Gamma j= #\Omega\Gamma n \Gamma 1; " 0 ): Proof. The results about the set theories are =-=from [5]-=-. ut 46 11 ACA 0 ` KT $ RFN \Pi 1 1 ((\Pi 1 2 \Gamma BI) 0 ) In view of Theorem 10.1 one is naturally led to search for a natural strengthening of (\Pi 1 2 \Gamma BI) 0 that proves WF (#\Omega ! ) and... |

1 |
Gallier:What's so special about Kruskal's theorem and the ordinal \Gamma 0 ? A survey of some results in proof theory. Annals of Pure and Applied Logic 53
- H
- 1991
(Show Context)
Citation Context ...Kruskal's theorem is especially felt due to the fact that this theorem figures prominently in computer science, because it is the main tool for showing that sets of rewrite rules are terminating (see =-=[3]-=-, p. 258, where this challenge is offered). Our paper gives a complete proof--theoretic characterization of Kruskal's theorem in terms of ordinal notation systems, subsystems of second order 1 arithme... |