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Prooftheoretic investigations on Kruskal's theorem
 Ann. Pure Appl. Logic
, 1993
"... In this paper we calibrate the exact prooftheoretic 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 [ ..."
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Cited by 23 (3 self)
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In this paper we calibrate the exact prooftheoretic 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 prooftheoretic 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 prooftheoretic 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 ...
An intuitionistic proof of Kruskal's Theorem
 Archive for Mathematical Logic
, 2000
"... this paper is to show that the arguments given by Higman and Kruskal are essentially constructive and acceptable from an intuitionistic point of view and that the later argument given by NashWilliams is not. The paper consists of the following 11 Sections. 1. Dickson's Lemma 2. Almost full relation ..."
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Cited by 8 (2 self)
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this paper is to show that the arguments given by Higman and Kruskal are essentially constructive and acceptable from an intuitionistic point of view and that the later argument given by NashWilliams is not. The paper consists of the following 11 Sections. 1. Dickson's Lemma 2. Almost full relations 3. Brouwer's Thesis 4. Ramsey's Theorem 5. The Finite Sequence Theorem 6. Vazsonyi's Conjecture for binary trees 7. Higman's Theorem 8. Vazsonyi's Conjecture and the Tree Theorem 9. MinimalBadSequence Arguments 10. The Principle of Open Induction 11. Concluding Remarks Except for Section 9, we will argue intuitionistically. 1 1 Dickson's Lemma
Worms, gaps and hydras
 Mathematical Logic Quarterly
, 2005
"... We define a direct translation from finite rooted trees to finite natural functions which shows that the Worm Principle introduced by Lev Beklemishev is equivalent to a very slight variant of the wellknown KirbyParis ’ Hydra Game. We further show that the elements in a reduction sequence of the Wo ..."
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Cited by 4 (2 self)
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We define a direct translation from finite rooted trees to finite natural functions which shows that the Worm Principle introduced by Lev Beklemishev is equivalent to a very slight variant of the wellknown KirbyParis ’ Hydra Game. We further show that the elements in a reduction sequence of the Worm Principle determine a bad sequence in the wellquasiordering of finite sequences of natural numbers with respect to Friedman’s gapembeddability. 1
An Inductive Version of NashWilliams’ MinimalBadSequence Argument for Higman’s Lemma
 In P. Callaghan, e.al., Types for Proofs and Programs, Lecture Notes in Computer Science 2277
, 2001
"... Abstract. Higman’s lemma has a very elegant, nonconstructive proof due to NashWilliams [NW63] using the socalled minimalbadsequence argument. The objective of the present paper is to give a proof that uses the same combinatorial idea, but is constructive. For a two letter alphabet this was done ..."
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Cited by 3 (1 self)
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Abstract. Higman’s lemma has a very elegant, nonconstructive proof due to NashWilliams [NW63] using the socalled minimalbadsequence argument. The objective of the present paper is to give a proof that uses the same combinatorial idea, but is constructive. For a two letter alphabet this was done by Coquand and Fridlender [CF94]. Here we present a proof in a theory of inductive definitions that works for arbitrary decidable well quasiorders. 1
Applications of inductive definitions and choice principles to program synthesis
"... Abstract. We describe two methods of extracting constructive content from classical proofs, focusing on theorems involving infinite sequences and nonconstructive choice principles. The first method removes any reference to infinite sequences and transforms the theorem into a system of inductive defi ..."
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Cited by 1 (0 self)
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Abstract. We describe two methods of extracting constructive content from classical proofs, focusing on theorems involving infinite sequences and nonconstructive choice principles. The first method removes any reference to infinite sequences and transforms the theorem into a system of inductive definitions, the other applies a combination of Gödel’s negativeand Friedman’s Atranslation. Both approaches are explained by means of a case study on Higman’s Lemma and its wellknown classical proof due to NashWilliams. We also discuss some prooftheoretic optimizations that were crucial for the formalization and implementation of this work in the interactive proof system Minlog. 1
Brief introduction to unprovability
"... Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. ..."
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Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. In addition, we obtain a short accessible proof of unprovability of the ParisHarrington Principle. The proof employs old ideas but uses only one colouring and directly extracts the set of indiscernibles from its homogeneous set. We also present modified, abridged statements whose unprovability proofs are especially simple. These proofs were tailored for teaching purposes. The article is intended to be accessible to the widest possible audience of mathematicians, philosophers and computer scientists as a brief survey of the subject, a guide through the literature in the field, an introduction to its modeltheoretic techniques and, finally, a modeltheoretic proof of a modern theorem in the subject. However, some understanding of logic is assumed on the part of the readers. The intended audience of this paper consists of logicians, logicaware mathematicians andthinkers of other backgrounds who are interested in unprovable mathematical statements.