Subsystems of Set Theory and Second Order Number Theory

Pohlers, Wolfram

doi:10.1016/s0049-237x(98)80019-0

Cited by 53 publications

(97 citation statements)

References 21 publications

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“…Characterizing the classes F(T ) for strong theories T in terms of transfinite hierarchies of recursive functions constitutes one of the main tasks of proof-theoretic ordinal analysis. See [68] for a survey of the modern developments in this important area of proof theory.…”

Section: Provably Total Computable Functions and Fragments Of Pamentioning

confidence: 99%

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Reflection principles and provability algebras in formal arithmetic

Beklemishev

2005

Russ. Math. Surv.

128

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Section: Provably Total Computable Functions and Fragments Of Pamentioning

confidence: 99%

“…(See, for example, a comprehensive survey [68].) These investigations, in particular, show that the "strength" of a sufficiently expressive theory can be characterized by certain countable ordinals.…”

Section: Introductionmentioning

confidence: 99%

Reflection principles and provability algebras in formal arithmetic

Beklemishev

2005

Russ. Math. Surv.

128

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“…Since these ideas have appeared in so many different contexts, often arising independently, trying to sort out the proper accreditations at each step along the way would be difficult; and so I hope this broad attribution is enough to acknowledge the general debt that this work owes to that which has come before. I should mention that I have also benefited a good deal from Buss' ordinal analysis of arithmetic, using the witnessing method, in [7]; from the emphasis on ordinal recursion and its properties, in Friedman and Sheard [11]; and, of course, from the traditional Gentzen-Schütte approach to ordinal analysis, surveyed in [21,22,24].…”

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confidence: 99%

“…In this paper I will focus on the upper bounds, but, in fact, all the upper bounds I provide will be sharp in this sense. (See [21,22] for more information on establishing the lower bounds.) §5.…”

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Ordinal analysis without proofs

Avigad¹

2017

Reflections on the Foundations of Mathematics

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Abstract. An approach to ordinal analysis is presented which is finitary, but highlights the semantic content of the theories under consideration, rather than the syntactic structure of their proofs. In this paper the methods are applied to the analysis of theories extending Peano arithmetic with transfinite induction and transfinite arithmetic hierarchies. §1. Introduction. As the name implies, in the field of proof theory one tends to focus on proofs. Nowhere is this emphasis more evident than in the field of ordinal analysis, where one typically designs procedures for "unwinding" derivations in appropriate deductive systems. One might wonder, however, if this emphasis is really necessary; after all, the results of an ordinal analysis describe a relationship between a system of ordinal notations and a theory, and it is natural to think of the latter as the set of semantic consequences of some axioms. From this point of view, it may seem disappointing that we have to choose a specific deductive system before we can begin the ordinal analysis.In fact, Hilbert's epsilon substitution method, historically the first attempt at finding a finitary consistency proof for arithmetic, has a more semantic character. With this method one uses so-called epsilon terms to reduce arithmetic to a quantifier-free calculus, and then one looks for a procedure that assigns numerical values to any finite set of closed terms, in a manner consistent with the axioms. The first ordinal analysis of arithmetic using epsilon terms is due to Ackermann [1]; for further developments see, for example, [20].2000 Mathematics Subject Classification. 03F15,03F05,03F30. Dedicated to Solomon Feferman on the occasion of his 70th birthday. It is an honor to be able to contribute a paper to this volume. Although I did my graduate work under Jack Silver at Berkeley, Sol served as an informal second advisor to me, and his thoughtful advice and guidance made frequent visits to Stanford both enjoyable and well worthwhile. Mathematical logic, as a discipline, is poised between philosophy and mathematics, and so has to answer to competing standards of philosophical relevance and mathematical elegance. Throughout his career, Sol has been able to strike a harmonious balance between the two, with work that is deeply satisfying on both counts. His style sets a high standard for future generations, and one that many of us will look to as a model.I would like to thank Lev Beklemishev for comments and suggestions, and the anonymous referees for their very careful readings.

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Kripke-Platek Set Theory and the Anti-Foundation Axiom

Rathjen

2001

MLQ - Math. Log. Quart.

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Subsystems of Set Theory and Second Order Number Theory

Cited by 53 publications

References 21 publications

Reflection principles and provability algebras in formal arithmetic

Reflection principles and provability algebras in formal arithmetic

Ordinal analysis without proofs

Kripke-Platek Set Theory and the Anti-Foundation Axiom

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