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Numeral completeness of weak theories of arithmetic.

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Title: Numeral completeness of weak theories of arithmetic.
Authors: Kahle, Reinhard1 (AUTHOR), Oitavem, Isabel2 (AUTHOR), Santos, Paulo Guilherme3 (AUTHOR)
Source: Journal of Logic & Computation. Jan2025, Vol. 35 Issue 1, p172-193. 22p.
Subjects: Incompleteness theorems, Arithmetic
Abstract: We study numeral forms of completeness and consistency for |$\mathsf {S}^1_2$| and other weak theories, like |$\mathsf {EA}$|⁠. This gives rise to an exploration of the derivability conditions needed to establish the mentioned results; a presentation of a weak form of Gödel's Second Incompleteness Theorem without using 'provability implies provable provability'; a provability predicate that satisfies the mentioned derivability condition for weak theories; and a completeness result via consistency statements. Moreover, the paper includes characterizations of the provability predicates for which the numeral results hold, having |$\mathsf {EA}$| as the surrounding theory, and results on functions that compute finitist consistency statements. [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: Numeral completeness of weak theories of arithmetic.
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  Data: <searchLink fieldCode="AR" term="%22Kahle%2C+Reinhard%22">Kahle, Reinhard</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Oitavem%2C+Isabel%22">Oitavem, Isabel</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Santos%2C+Paulo+Guilherme%22">Santos, Paulo Guilherme</searchLink><relatesTo>3</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Logic+%26+Computation%22">Journal of Logic & Computation</searchLink>. Jan2025, Vol. 35 Issue 1, p172-193. 22p.
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  Data: We study numeral forms of completeness and consistency for |$\mathsf {S}^1_2$| and other weak theories, like |$\mathsf {EA}$|⁠. This gives rise to an exploration of the derivability conditions needed to establish the mentioned results; a presentation of a weak form of Gödel's Second Incompleteness Theorem without using 'provability implies provable provability'; a provability predicate that satisfies the mentioned derivability condition for weak theories; and a completeness result via consistency statements. Moreover, the paper includes characterizations of the provability predicates for which the numeral results hold, having |$\mathsf {EA}$| as the surrounding theory, and results on functions that compute finitist consistency statements. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Group: Ab
  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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    Identifiers:
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        Value: 10.1093/logcom/exad075
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      – Code: eng
        Text: English
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        PageCount: 22
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      – SubjectFull: Incompleteness theorems
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      – SubjectFull: Arithmetic
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      – TitleFull: Numeral completeness of weak theories of arithmetic.
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            NameFull: Kahle, Reinhard
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            NameFull: Oitavem, Isabel
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            NameFull: Santos, Paulo Guilherme
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              Text: Jan2025
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              Y: 2025
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