Serial properties, selector proofs and the provability of consistency.
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| Title: | Serial properties, selector proofs and the provability of consistency. |
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| 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}}$| in PA. These findings show that PA proves its consistency whereas, by Gödel's second incompleteness theorem, PA cannot prove its uniform consistency. [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
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| Abstract: | The consistency of a theory means that each of its formal derivations |$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] |
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| ISSN: | 0955792X |
| DOI: | 10.1093/logcom/exae034 |
