# Über das Verhältnis zwischen intuitionistischer und klassischer Arithmetik

@article{Gentzen1974berDV, title={{\"U}ber das Verh{\"a}ltnis zwischen intuitionistischer und klassischer Arithmetik}, author={Gerhard Gentzen}, journal={Archiv f{\"u}r mathematische Logik und Grundlagenforschung}, year={1974}, volume={16}, pages={119-132} }

## 37 Citations

Interpolation via translations

- Computer ScienceMath. Log. Q.
- 2009

A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This…

A Lightweight Double-negation Translation

- Computer ScienceLPAR
- 2015

A partial order among translations is presented to select translations avoiding as many double negations as possible, and this order admits a unique minimal syntax-directed translation with noticeable properties.

Double Negation Semantics for Generalisations of Heyting Algebras

- Computer Science, MathematicsStud Logica
- 2021

An algebraic framework for investigating proposed translations of classical logic into intuitionistic logic, such as the four negative translations introduced by Kolmogorov, Gödel, Gentzen and Glivenko, and proves that the syntactic version of these translations will fail to satisfy Troelstra’s criteria in the corresponding substructural logical setting.

On Correspondence between Selective CPS Transformation and Selective Double Negation Translation

- Computer Science
- 2021

This paper provides a logical account of a CBV selective CPS transformation by defining a selective DNT via the Curry-Howard isomorphism, and uses an annotated proof system derived from the corresponding type and effect system to translate classical proofs into equivalent intuitionistic proofs, smaller than those obtained by the usual DNTs.

A Mathematical Model of Quantum Computer by Both Arithmetic and Set Theory

- Computer Science
- 2020

A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of the authors' computers, and that formal relation of “reality’, “representation”, and “language” modeled by (at least two) Turing machines is investigated.

The quantum strategy of completeness: On the self-foundation of mathematics

- Mathematics
- 2020

Gentzen’s approach by transfinite induction and that of intuitionist Heyting arithmetic to completeness and the self-foundation of mathematics are compared and opposed to the Gödel incompleteness…

Cut elimination for entailment relations

- Computer Science, MathematicsArch. Math. Log.
- 2019

This paper state and prove a cut-elimination result for inductively generated entailment relations and analyse some of its consequences and describe the existing connections with analogous results in the literature.

Negative Translations for Affine and Lukasiewicz Logic

- Mathematics, Computer ScienceArXiv
- 2019

It is proved that a wide class of formula translations including those of Kolmogorov, Godel, Gentzen and Glivenko not only satisfy Troelstra's criteria with respect to a natural intuitionistic fragment of Lukasiewicz logic but are all equivalent.

On Relating Theories: Proof-Theoretical Reduction

- MathematicsMathesis Universalis, Computability and Proof
- 2019

The notion of proof-theoretical or finitistic reduction of one theory to another has a long tradition. Feferman and Sieg (Buchholz et al., Iterated inductive definitions and subsystems of analysis.…

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