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Serial properties, selector proofs and the provability of consistency.

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Title: Serial properties, selector proofs and the provability of consistency.
Authors: Artemov, Sergei1 (AUTHOR)
Source: Journal of Logic & Computation. Apr2025, Vol. 35 Issue 3, p1-18. 18p.
Subjects: Incompleteness theorems, Arithmetic, Contradiction
Abstract: The consistency of a theory means that each of its formal derivations |$D_{0}, D_{1}, D_{2}, \ldots $| is free of contradictions. For Peano Arithmetic PA , after the standard coding of derivations by numerals, PA -consistency is directly represented by the consistency scheme |$\textsf{Con}^{S}_{\textsf{PA}}$|⁠ , which is a series of arithmetical statements ' |$n$| is not a code of a derivation of |$\ (0=1)$| ' for numerals |$n=0,1,2,\ldots $|⁠. We note that the consistency formula |$\textsf{Con}_{\textsf{PA}}$|⁠ , |$\forall x$| ' |$x$| is not a code of a derivation of |$(0=1)$|⁠ , ' is strictly stronger in PA than PA -consistency and corresponds to some other property, which we call uniform consistency. When studying the provability of consistency in PA we ought to work not with the consistency formula |$\textsf{Con}_{\textsf{PA}}$| but rather with the consistency scheme |$\textsf{Con}^{S}_{\textsf{PA}}$|⁠ , which adequately represents PA -consistency. This paper introduces the Hilbert-inspired notion of proof of an infinite series of formulas in a theory and proves PA -consistency in the form |$\textsf{Con}^{S}_{\textsf{PA}}$| in PA. These findings show that PA proves its consistency whereas, by Gödel's second incompleteness theorem, PA cannot prove its uniform consistency. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Logic & Computation is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Serial properties, selector proofs and the provability of consistency.
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Logic+%26+Computation%22">Journal of Logic & Computation</searchLink>. Apr2025, Vol. 35 Issue 3, p1-18. 18p.
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  Data: <searchLink fieldCode="DE" term="%22Incompleteness+theorems%22">Incompleteness theorems</searchLink><br /><searchLink fieldCode="DE" term="%22Arithmetic%22">Arithmetic</searchLink><br /><searchLink fieldCode="DE" term="%22Contradiction%22">Contradiction</searchLink>
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  Data: The consistency of a theory means that each of its formal derivations |$D_{0}, D_{1}, D_{2}, \ldots $| is free of contradictions. For Peano Arithmetic PA , after the standard coding of derivations by numerals, PA -consistency is directly represented by the consistency scheme |$\textsf{Con}^{S}_{\textsf{PA}}$|⁠ , which is a series of arithmetical statements ' |$n$| is not a code of a derivation of |$\ (0=1)$| ' for numerals |$n=0,1,2,\ldots $|⁠. We note that the consistency formula |$\textsf{Con}_{\textsf{PA}}$|⁠ , |$\forall x$| ' |$x$| is not a code of a derivation of |$(0=1)$|⁠ , ' is strictly stronger in PA than PA -consistency and corresponds to some other property, which we call uniform consistency. When studying the provability of consistency in PA we ought to work not with the consistency formula |$\textsf{Con}_{\textsf{PA}}$| but rather with the consistency scheme |$\textsf{Con}^{S}_{\textsf{PA}}$|⁠ , which adequately represents PA -consistency. This paper introduces the Hilbert-inspired notion of proof of an infinite series of formulas in a theory and proves PA -consistency in the form |$\textsf{Con}^{S}_{\textsf{PA}}$| in PA. These findings show that PA proves its consistency whereas, by Gödel's second incompleteness theorem, PA cannot prove its uniform consistency. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Journal of Logic & Computation is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1093/logcom/exae034
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        Text: English
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      – SubjectFull: Incompleteness theorems
        Type: general
      – SubjectFull: Arithmetic
        Type: general
      – SubjectFull: Contradiction
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      – TitleFull: Serial properties, selector proofs and the provability of consistency.
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              Text: Apr2025
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              Y: 2025
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