Proof Theory

Schütte, Kurt

doi:10.1007/978-3-642-66473-1

Cited by 284 publications

(101 citation statements)

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“…Now instead of using indirect systems of (numerical) notations for ordinals, it would be much more natural to use terms of our systems which provably denote in them von Neumann's ordinals. We expect that every ordinal less than Γ 0 , the Feferman-Schütte ordinal for predicativity ( [15,17,44,45]), should be obtainable in this way. Decoding: Although {ω n | n ∈ N} and λn ∈ N.ω n are not definable in P ZF , { ω n | n ∈ N} and λn ∈ N. ω n are definable, where ω n is some natural Gödel code in HF for the term of L P ZF that defines ω n .…”

Section: Strengthening P Zfmentioning

confidence: 99%

A New Approach to Predicative Set Theory

Avron¹

2010

Ways of Proof Theory

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We suggest a new basic framework for the Weyl-Feferman predicativist program by constructing a formal predicative set theory P ZF which resembles ZF . The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the "surrounding universe". This idea is implemented using syntactic safety relations between formulas and sets of variables. These safety relations generalize both the notion of domain-independence from database theory, and Godel notion of absoluteness from set theory. The language of P ZF is type-free, and it reflects real mathematical practice in making an extensive use of statically defined abstract set terms. Another important feature of P ZF is that its underlying logic is ancestral logic (i.e. the extension of FOL with a transitive closure operation).

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Section: Strengthening P Zfmentioning

confidence: 99%

A New Approach to Predicative Set Theory

Avron¹

2010

Ways of Proof Theory

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“…We assume that the reader is familiar with the basic ordinal theory, the Veblen hierarchy of normal functions and collapsing functions à la Buchholz. A full exposition can be found in Buchholz and Schütte [8], Pohlers [23,24], and Schütte [25].…”

Section: Ordinal Notationsmentioning

confidence: 99%

“…Now we set α = NF ϕβγ :⇐⇒ α = ϕβγ and β, γ < α and obtain the following normal form property. For a proof see, for example, Pohlers [23] or Schütte [25].…”

Section: Ordinal Notationsmentioning

confidence: 99%

Theories of Proof-Theoretic Strength Ψ (Γ_Ω +1)

Buchholtz¹,

Jäger²,

Strahm³

2016

Concepts of Proof in Mathematics, Philosophy, and Computer Science

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The purpose of this article is to present a range of theories with proof-theoretic ordinal ψ(Γ Ω+1 ). This ordinal parallels the ordinal of predicative analysis, Γ 0 , and our theories are parallel to classical theories of strength Γ 0 such as ID <ω , FP 0 , ATR 0 , Σ 1 1 -DC 0 + (SUB), and Σ 1 1 -AC 0 + (SUB). We also relate these theories to the unfolding of ID 1 which was already presented in the PhD thesis of the first author as a system of strength ψ(Γ Ω+1 ).

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“…FP-ACA* is infinitary with respect to the ranks of its formulas and the length of its derivations and is related to the system RA* of Schiitte [12].…”

Section: Arithmetic Comprehension Ac a For All Arithmetic Formumentioning

confidence: 99%

“…However, the principal formulas of the fixed point rules have rank 0. Therefore by applying the standard techniques of predicative proof theory as developed for example in Girard [6], Schiitte [12], or Takeuti [14], one obtains the following weaker result. …”

Section: R=>af(s) For Alls Rf(s)=>a For Some S R => A(vx)f(x) ' Rmentioning

confidence: 99%

About the proof-theoretic ordinals of weak fixed point theories

Jäger¹,

Primo

1992

J. symb. log.

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Abstract. This paper presents several proof-theoretic results concerning weak fixed point theories over second order number theory with arithmetic comprehension and full or restricted induction on the natural numbers. It is also shown that there are natural second order theories which are proof-theoretically equivalent but have different proof-theoretic ordinals.Fixed points and fixed point theories play an important role in many branches of mathematical logic and theoretical computer science. The spectrum ranges from the fixed point theorem in recursion theory to fixed point arguments in categorical logic and denotational semantics of high-level programming languages.In proof theory, special emphasis has been put on formal systems for fixed points of (iterated) inductive definitions and their relationship to subsystems of analysis, set theory and constructive mathematics (cf. e.g. Buchholz, Feferman, Pohlers and Sieg [2], Feferman [3], and Jager [7]). However, interpreted in the proper sense, also features of modern type theories can be studied in terms of fixed points, and many concepts in nonmonotonic reasoning (circumscription, completion of theories, etc.) are related to fixed point theories. To a certain extent even parts of logic programming are built upon fixed point constructions (cf. e.g. Lloyd [9] and Jager and Stark [8]).The general purpose of this paper is to study several proof-theoretic aspects of the fixed point theories FP-ACA 0 and FP-ACA. They are formulated in the language of second order arithmetic, contain the axioms of primitive recursive arithmetic PRA, and comprise comprehension for arithmetic formulas. In addition, there has to be a fixed point for every positive arithmetic definition clause, and this fixed point can be proved to define a set. Both theories differ in the principles of complete induction which are available: In FP-ACA 0 complete induction on the natural numbers is restricted to sets, whereas FP-ACA contains complete induction for arbitrary formulas.We will show that the proof-theoretic ordinals of FP-ACA 0 and FP-ACA are £ 0 and

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Proof Theory

Cited by 284 publications

References 0 publications

A New Approach to Predicative Set Theory

A New Approach to Predicative Set Theory

Theories of Proof-Theoretic Strength Ψ (Γ_Ω +1)

About the proof-theoretic ordinals of weak fixed point theories

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Proof Theory

Cited by 284 publications

References 0 publications

A New Approach to Predicative Set Theory

A New Approach to Predicative Set Theory

Theories of Proof-Theoretic Strength Ψ (ΓΩ +1)

About the proof-theoretic ordinals of weak fixed point theories

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Product

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Theories of Proof-Theoretic Strength Ψ (Γ_Ω +1)