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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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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.
Algorithms for ordinal arithmetic
 In 19th International Conference on Automated Deduction (CADE
, 2003
"... Abstract. Proofs of termination are essential for establishing the correct behavior of computing systems. There are various ways of establishing termination, but the most general involves the use of ordinals. An example of a theorem proving system in which ordinals are used to prove termination is A ..."
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Abstract. Proofs of termination are essential for establishing the correct behavior of computing systems. There are various ways of establishing termination, but the most general involves the use of ordinals. An example of a theorem proving system in which ordinals are used to prove termination is ACL2. In ACL2, every function defined must be shown to terminate using the ordinals up to ɛ0. We use a compact notation for the ordinals up to ɛ0 (exponentially more succinct than the one used by ACL2) and define efficient algorithms for ordinal addition, subtraction, multiplication, and exponentiation. In this paper we describe our notation and algorithms, prove their correctness, and analyze their complexity. 1
Ordinal arithmetic: Algorithms and mechanization
 Journal of Automated Reasoning
, 2006
"... Abstract. Termination proofs are of critical importance for establishing the correct behavior of both transformational and reactive computing systems. A general setting for establishing termination proofs involves the use of the ordinal numbers, an extension of the natural numbers into the transfini ..."
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Abstract. Termination proofs are of critical importance for establishing the correct behavior of both transformational and reactive computing systems. A general setting for establishing termination proofs involves the use of the ordinal numbers, an extension of the natural numbers into the transfinite which were introduced by Cantor in the nineteenth century and are at the core of modern set theory. We present the first comprehensive treatment of ordinal arithmetic on compact ordinal notations and give efficient algorithms for various operations, including addition, subtraction, multiplication, and exponentiation. Using the ACL2 theorem proving system, we implemented our ordinal arithmetic algorithms, mechanically verified their correctness, and developed a library of theorems that can be used to significantly automate reasoning involving the ordinals. To enable users of the ACL2 system to fully utilize our work required that we modify ACL2, e.g., we replaced the underlying representation of the ordinals and added a large library of definitions and theorems. Our modifications are available starting with ACL2 version 2.8. 1.
The Higher Infinite in Proof Theory
 Logic Colloquium '95. Lecture Notes in Logic
, 1995
"... this paper. The exposition here diverges from the presentation given at the conference in two regards. Firstly, the talk began with a broad introduction, explaining the current rationale and goals of ordinaltheoretic proof theory, which take the place of the original Hilbert Program. Since this par ..."
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this paper. The exposition here diverges from the presentation given at the conference in two regards. Firstly, the talk began with a broad introduction, explaining the current rationale and goals of ordinaltheoretic proof theory, which take the place of the original Hilbert Program. Since this part of the talk is now incorporated in the first two sections of the BSLpaper [48] there is no point in reproducing it here. Secondly, we shall omit those parts of the talk concerned with infinitary proof systems of ramified set theory as they can also be found in [48] and even more detailed in [45]. Thirdly, thanks to the aforementioned omissions, the advantage of present paper over the talk is to allow for a much more detailed account of the actual information furnished by ordinal analyses and the role of large cardinal hypotheses in devising ordinal representation systems. 2 Observations on ordinal analyses
A Comparison of Two Systems of Ordinal Notations
"... I show how the Bachmann method of generating countable ordinals using uncountable ordinals can be replaced by the use of higher order xed point extractors available in the term calculus of Howard's system of constructive ordinals. This leads to a notion of the intrinsic complexity of a notated ordin ..."
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I show how the Bachmann method of generating countable ordinals using uncountable ordinals can be replaced by the use of higher order xed point extractors available in the term calculus of Howard's system of constructive ordinals. This leads to a notion of the intrinsic complexity of a notated ordinal analogous to the intrinsic complexity of a numeric function described in Gödel's T.
THE VEBLEN FUNCTIONS FOR COMPUTABILITY THEORISTS
, 2010
"... We study the computabilitytheoretic complexity and prooftheoretic strength of the following statements: (1) “If X is a wellordering, then so is εX”, and (2) “If X is a wellordering, then so is ϕ(α, X)”, where α is a fixed computable ordinal and ϕ represents the twoplaced Veblen function. For th ..."
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We study the computabilitytheoretic complexity and prooftheoretic strength of the following statements: (1) “If X is a wellordering, then so is εX”, and (2) “If X is a wellordering, then so is ϕ(α, X)”, where α is a fixed computable ordinal and ϕ represents the twoplaced Veblen function. For the former statement, we show that ω iterations of the Turing jump are necessary in the proof and that the statement is equivalent to ACA + 0 over RCA0. To prove the latter statement we need to use ωα iterations of the Turing jump, and we show that the statement is equivalent to Π0 ωαCA0. Our proofs are purely computabilitytheoretic. We also give a new proof of a result of Friedman: the statement “if X is a wellordering, then so is ϕ(X, 0)” is equivalent to ATR0 over RCA0.
Covering of ordinals
 LIPICS LEIBNIZ INTERNATIONAL PROCEEDINGS IN INFORMATICS
, 2009
"... The paper focuses on the structure of fundamental sequences of ordinals smaller than ε0. A first result is the construction of a monadic secondorder formula identifying a given structure, whereas such a formula cannot exist for ordinals themselves. The structures are precisely classified in the pu ..."
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The paper focuses on the structure of fundamental sequences of ordinals smaller than ε0. A first result is the construction of a monadic secondorder formula identifying a given structure, whereas such a formula cannot exist for ordinals themselves. The structures are precisely classified in the pushdown hierarchy. Ordinals are also located in the hierarchy, and a direct presentation is given.
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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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.