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On the Constructive Truth and Falsity in Peano Arithmetic

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Logical Foundations of Computer Science (LFCS 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11972))

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Abstract

Recently, Artemov [4] offered the notion of constructive truth and falsity in the spirit of Brouwer-Heyting-Kolmogorov semantics and its formalization, the Logic of Proofs. In this paper, we provide a complete description of constructive truth and falsity for Friedman’s constant fragment of Peano Arithmetic. For this purpose, we generalize the constructive falsity to n-constructive falsity where n is any positive natural number. We also establish similar classification results for constructive truth and n-constructive falsity of Friedman’s formulas. Then, we discuss ‘extremely’ independent sentences in the sense that they are classically true but neither constructively true nor n-constructive false for any n.

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Notes

  1. 1.

    A detailed proof of the provable \(\varSigma _1\)-completeness is found on pp. 46–49 of [11].

  2. 2.

    As usual, PA\(^n\) is defined: PA\(^1\)= PA; PA\(^{n+1}= \) PA\(^n + Con(\textsf {PA}^n)\).

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Author information

Authors and Affiliations

  1. Computer Science, Graduate Center, City University of New York, New York, USA

    Hirohiko Kushida

Authors
  1. Hirohiko Kushida
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Correspondence to Hirohiko Kushida .

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Editors and Affiliations

  1. The Graduate Center, CUNY, New York, NY, USA

    Sergei Artemov

  2. Cornell University, Ithaca, NY, USA

    Anil Nerode

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Kushida, H. (2020). On the Constructive Truth and Falsity in Peano Arithmetic. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2020. Lecture Notes in Computer Science(), vol 11972. Springer, Cham. https://doi.org/10.1007/978-3-030-36755-8_5

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