Proof Theory

doi:10.1007/978-3-540-69319-2

Cited by 22 publications

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References 43 publications

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“…In fact, not much ordinal theory is presumed in this article; we almost exclusively work with ordinals less than ϕ 2 0, which can be written in terms of addition and the functions ξ → ω ξ , ξ → ε ξ . For more details, we refer the reader to standard texts such as [10,12].…”

Section: Basic Definitionsmentioning

confidence: 99%

“…ACA 0 asserts that, for all n ∈ N and X ⊆ N, the n-Turing jump of X exists, while ACA + 0 asserts that its ω-jump exists; see [13] for details. The proof-theoretic ordinal of ACA 0 is ε ω [1], and that of ACA + 0 is ϕ 2 (0) [9]; we will briefly review these ordinals later in the text, but refer the reader to standard texts such as [10,12] for a more detailed treatment of proof-theoretic ordinals.…”

Section: Introductionmentioning

confidence: 99%

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Ackermannian Goodstein Sequences of Intermediate Growth

Fernández–Duque

Weiermann

2020

Lecture Notes in Computer Science

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The original Goodstein process proceeds by writing natural numbers in nested exponential k-normal form, then successively raising the base to k +1 and subtracting one from the end result. Such sequences always reach zero, but this fact is unprovable in Peano arithmetic. In this paper we instead consider notations for natural numbers based on the Ackermann function. We define two new Goodstein processes, obtaining new independence results for ACA 0 and ACA + 0 , theories of second order arithmetic related to the existence of Turing jumps.

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Section: Basic Definitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Ackermannian Goodstein Sequences of Intermediate Growth

Fernández–Duque

Weiermann

2020

Lecture Notes in Computer Science

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A Compositional Theory of Krivine’s Classical Realisability

Hayashi,

Leigh

2024

Lecture Notes in Computer Science

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An Introduction to Proof Theory

Mancosu

Galvan

Zach

2021

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Proof theory is a central area of mathematical logic of special interest to philosophy. It has its roots in the foundational debate of the 1920s, in particular, in Hilbert’s program in the philosophy of mathematics, which called for a formalization of mathematics, as well as for a proof, using philosophically unproblematic, “finitary” means, that these systems are free from contradiction. Structural proof theory investigates the structure and properties of proofs in different formal deductive systems, including axiomatic derivations, natural deduction, and the sequent calculus. Central results in structural proof theory are the normalization theorem for natural deduction, proved here for both intuitionistic and classical logic, and the cut-elimination theorem for the sequent calculus. In formal systems of number theory formulated in the sequent calculus, the induction rule plays a central role. It can be eliminated from proofs of sequents of a certain elementary form: every proof of an atomic sequent can be transformed into a “simple” proof. This is Hilbert’s central idea for giving finitary consistency proofs. The proof requires a measure of proof complexity called an ordinal notation. The branch of proof theory dealing with mathematical systems such as arithmetic thus has come to be called ordinal proof theory. The theory of ordinal notations is developed here in purely combinatorial terms, and the consistency proof for arithmetic presented in detail.

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Proof Theory

Cited by 22 publications

References 43 publications

Ackermannian Goodstein Sequences of Intermediate Growth

Ackermannian Goodstein Sequences of Intermediate Growth

A Compositional Theory of Krivine’s Classical Realisability

An Introduction to Proof Theory

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