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What's so special about Kruskal's Theorem AND THE ORDINAL Γ0? A SURVEY OF SOME RESULTS IN PROOF THEORY
 ANNALS OF PURE AND APPLIED LOGIC, 53 (1991), 199260
, 1991
"... This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Kruskal’s tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, an ..."
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Cited by 43 (3 self)
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This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Kruskal’s tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Veblen hierarchies, some subsystems of secondorder logic, slowgrowing and fastgrowing hierarchies including Girard’s result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to Kruskal, the “tree theorem”, as well as a “finite miniaturization ” of Kruskal’s theorem due to Harvey Friedman. These versions of Kruskal’s theorem are remarkable from a prooftheoretic point of view because they are not provable in relatively strong logical systems. They are examples of socalled “natural independence phenomena”, which are considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Gödel. Kruskal’s tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of KnuthBendix completion procedures. There is also a close connection between a certain infinite countable ordinal called Γ0 and Kruskal’s theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence.
The Mathematical Import Of Zermelo's WellOrdering Theorem
 Bull. Symbolic Logic
, 1997
"... this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs ..."
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Cited by 5 (1 self)
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this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his wellknown paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions f :
Generating ordinal notations from below with a nonrecursive construction of the Schütte brackets
"... hsimmons @ manchester.ac.uk I rework Veblen’s ideas to show how hierarchies of normal functions can be generated merely by iterating certain higher order gadgets. As an illustration I show that an application of a Schütte bracket can be evaluated without the need of an intricate recursion. ..."
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Cited by 1 (1 self)
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hsimmons @ manchester.ac.uk I rework Veblen’s ideas to show how hierarchies of normal functions can be generated merely by iterating certain higher order gadgets. As an illustration I show that an application of a Schütte bracket can be evaluated without the need of an intricate recursion.
A.Miller Long Borel Hierarchies 1 Long Borel
, 704
"... We show that it is relatively consistent with ZF that the Borel hierarchy on the reals has length ω2. This implies that ω1 has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has length any given limit ..."
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We show that it is relatively consistent with ZF that the Borel hierarchy on the reals has length ω2. This implies that ω1 has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has length any given limit ordinal less than ω2, e.g., ω or ω1 + ω1.
Contents
, 2013
"... • A brief history of proof theory • Sequent calculi for classical and intuitionistic logic, Gentzen’s Hauptsatz: Cut elimination ..."
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• A brief history of proof theory • Sequent calculi for classical and intuitionistic logic, Gentzen’s Hauptsatz: Cut elimination
possible and for his help with this paper.
"... In the year before Zermelo published his proof of the wellordering principle from AC, the renowned Cambridge mathematician G. H. Hardy published a proof that there is an uncountable wellorderable subset of the real line [1], [2]. Hardy’s technique was surprisingly modern: he used a ladder system o ..."
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In the year before Zermelo published his proof of the wellordering principle from AC, the renowned Cambridge mathematician G. H. Hardy published a proof that there is an uncountable wellorderable subset of the real line [1], [2]. Hardy’s technique was surprisingly modern: he used a ladder system on ω1 to build an uncountable set of sequences of natural numbers wellordered by the preorder < ∗ of eventual domination. We now know that some form of the axiom of choice (AC) is needed for this, and even that the existence of Hardy’s uncountable set does not imply the existence of a ladder system on ω1 just assuming ZF. Definition 1. Given a limit ordinal α, a ladder at α is a strictly ascending sequence of ordinals less than α whose supremum is α. Given an ordinal γ, a ladder system on γ is a family {Lα: α ∈ γ, α is a limit ordinal of countable cofinality} where each Lα is a ladder at α. Theorem 1. In ZF, the following axioms are equivalent: 1. There is a ladder system on ω1. 2. There is a Hausdorff gap. 3a. There is a wellorderable special Aronszajn tree. 3b. There is a wellorderable special tree of height ω1. 3c. There is an Rspecial tree T of height ω1 with a choice function for the levels of