Results 51 to 60 of about 86 (68)
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From Second-Order Heyting Arithmetic to Second-Order Peano Arithmetic
1998Second-Order Peano Arithmetic (2PA) is obtained from Peano Arithmetic (PA) by adding second-order quantifiers, ∀2 and ∃2, and atomic formulae of the form N ⊨ α. In this chapter I shall extend the interpretation of PA in HA given in Chapter 33 to an interpretation of 2PA in 2HA.
Fletcher Peter
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Interpretations of Heyting's arithmetic—An analysis by means of a language with set symbols
Annals of Mathematical Logic, 1980 AbstractWell-known interpretations of Heyting's arithmetic of all finite types are the Diller-Nahm λ-interpretation [1] and Kreisel's modified realizability, subsequently called mr-interpretation [4]. For both interpretations one can define hybrids λ q resp.
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From Logic of Partial Terms to Heyting Arithmetic
1998Heyting Arithmetic (HA) is first-order intuitionistic number theory. It will be obtained from the Logic of Partial Terms (LPT) by restricting the range of the variables to numbers. Every HA derivation can be transformed into an LPT derivation. Therefore every theorem of HA (without free variables) has an intuitionistic proof.
Fletcher Peter
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On Properties of Feasibility in Non-Standard Heyting Arithmetic
2024 IEEE 3rd Conference on Information Technology and Data Science (CITDS)Peter Battyányi
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Note on extensions of Heyting's arithmetic by adding the "creative subject"
Archive for Mathematical Logic, 1999It is shown that the extension of Heyting's arithmetic HA by Kreisel's axioms for the creative subject [see \textit{A. S. Troelstra} and \textit{D. van Dalen}, Constructivism in mathematics. Vol. 1 (North-Holland, Amsterdam) (1988; Zbl 0653.03040), p. 236], with the induction schema restricted to arithmetical formulae, is conservative over HA.
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Algorithmic Equality in Heyting Arithmetic Modulo
2008Deduction Modulo is a formalism that aims at distinguish reasoning from computation in proofs. A theory modulo is formed with a set of axioms and a congruence defined by rewrite rules: the reasoning part of the theory is given by the axioms, the computational part by the congruence.
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From Second-Order Logic of Partial Terms to Second-Order Heyting Arithmetic
1998Second-Order Heyting Arithmetic (2HA) is obtained from Heyting Arithmetic (HA) by admitting atomic formulae of the form N ⊨ α and the second-order universal quantifier. In this chapter I shall extend the interpretation of HA in LPT given in Chapter 31 to an interpretation of 2HA in 2LPT.
Fletcher Peter
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Interactive Learning-Based Realizability Interpretation for Heyting Arithmetic with EM 1
2009We interpret classical proofs as constructive proofs (with constructive rules for *** , ***) over a suitable structure ${\mathcal N}$ for the language of natural numbers and maps of Godel's system ${\mathcal{T}}$. We introduce a new Realization semantics we call "Interactive learning-based Realizability", for Heyting Arithmetic plus EM 1 (Excluded ...
Federico Aschieri, Stefano Berardi
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On the Realizability of Prime Conjectures in Heyting Arithmetic
CoRR28 pages, 6 figures. Integrates constructive arithmetic, realizability semantics, and geometric logic to analyze the logical boundary of primality in context of the arithmetical ...
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A Direct Gentzen-Style Consistency Proof for Heyting Arithmetic
2015Gerhard Gentzen was the first to give a proof of the consistency of Peano Arithmetic and in all he worked out four different proofs between 1934 and 1939. The second proof was published as [1], the third as [2], and the fourth as [3]. The first proof was published posthumously in English translation in [4] and in the German original as [5].
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