A new perspective on completeness and finitist consistency.
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| Title: | A new perspective on completeness and finitist consistency. |
|---|---|
| Authors: | Santos, Paulo Guilherme1 (AUTHOR), Sieg, Wilfried2 (AUTHOR), Kahle, Reinhard3 (AUTHOR) |
| Source: | Journal of Logic & Computation. Sep2024, Vol. 34 Issue 6, p1179-1198. 20p. |
| Subjects: | Incompleteness theorems, Recursive functions, Number systems, Axioms |
| Abstract: | In this paper, we study the metamathematics of consistent arithmetical theories |$T$| (containing |$\textsf {I}\varSigma _{1}$|); we investigate numerical properties based on proof predicates that depend on numerations of the axioms. Numeral Completeness. For every true (in |$\mathbb {N}$|) sentence |$\vec {Q}\vec {x}.\varphi (\vec {x})$| , with |$\varphi (\vec {x})$| a |$\varSigma _{1}(\textsf {I}\varSigma _1)$| -formula, there is a numeration |$\tau $| of the axioms of |$T$| such that |$\textsf {I}\varSigma _1\vdash \vec {Q}\vec {x}. \texttt {Pr}_{\tau }(\ulcorner \varphi (\overset {\text{.} }{\vec {x}})\urcorner)$| , where |$\texttt {Pr}_{\tau }$| is the provability predicate for the numeration |$\tau $|. Numeral Consistency. If |$T$| is consistent, there is a |$\varSigma _{1}(\textsf {I}\varSigma _1)$| -numeration |$\tau $| of the axioms of |$\textsf {I}\varSigma _{1}$| such that |$\textsf {I}\varSigma _1\vdash \forall\, x. \texttt {Pr}_{\tau }(\ulcorner \neg \textit {Prf}(\ulcorner \perp \urcorner , \overset {\text{.}}{x})\urcorner)$| , where |$\textit {Prf}(x,y)$| denotes a |$\varDelta _{1}(\textsf {I}\varSigma _1)$| -definition of ' |$y$| is a |$T$| -proof of |$x$| '. Finitist consistency is addressed by generalizing a result of Artemov: Partial finitism. If |$T$| is consistent, there is a primitive recursive function |$f$| such that, for all |$n\in \mathbb {N}$| , |$f(n)$| is the code of an |$\textsf {I}\varSigma _{1}$| -proof of |$\neg\, \textit{Prf}(\ulcorner \perp \urcorner ,\overline {n})$|. These results are not in conflict with Gödel's Incompleteness Theorems. Rather, they allow to extend their usual interpretation and show a deep connection to reflections in Hilbert's last papers of 1931. [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.) | |
| Database: | Engineering Source |
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| Header | DbId: egs DbLabel: Engineering Source An: 179665018 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: A new perspective on completeness and finitist consistency. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Santos%2C+Paulo+Guilherme%22">Santos, Paulo Guilherme</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Sieg%2C+Wilfried%22">Sieg, Wilfried</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Kahle%2C+Reinhard%22">Kahle, Reinhard</searchLink><relatesTo>3</relatesTo> (AUTHOR) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Logic+%26+Computation%22">Journal of Logic & Computation</searchLink>. Sep2024, Vol. 34 Issue 6, p1179-1198. 20p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Incompleteness+theorems%22">Incompleteness theorems</searchLink><br /><searchLink fieldCode="DE" term="%22Recursive+functions%22">Recursive functions</searchLink><br /><searchLink fieldCode="DE" term="%22Number+systems%22">Number systems</searchLink><br /><searchLink fieldCode="DE" term="%22Axioms%22">Axioms</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: In this paper, we study the metamathematics of consistent arithmetical theories |$T$| (containing |$\textsf {I}\varSigma _{1}$|); we investigate numerical properties based on proof predicates that depend on numerations of the axioms. Numeral Completeness. For every true (in |$\mathbb {N}$|) sentence |$\vec {Q}\vec {x}.\varphi (\vec {x})$| , with |$\varphi (\vec {x})$| a |$\varSigma _{1}(\textsf {I}\varSigma _1)$| -formula, there is a numeration |$\tau $| of the axioms of |$T$| such that |$\textsf {I}\varSigma _1\vdash \vec {Q}\vec {x}. \texttt {Pr}_{\tau }(\ulcorner \varphi (\overset {\text{.} }{\vec {x}})\urcorner)$| , where |$\texttt {Pr}_{\tau }$| is the provability predicate for the numeration |$\tau $|. Numeral Consistency. If |$T$| is consistent, there is a |$\varSigma _{1}(\textsf {I}\varSigma _1)$| -numeration |$\tau $| of the axioms of |$\textsf {I}\varSigma _{1}$| such that |$\textsf {I}\varSigma _1\vdash \forall\, x. \texttt {Pr}_{\tau }(\ulcorner \neg \textit {Prf}(\ulcorner \perp \urcorner , \overset {\text{.}}{x})\urcorner)$| , where |$\textit {Prf}(x,y)$| denotes a |$\varDelta _{1}(\textsf {I}\varSigma _1)$| -definition of ' |$y$| is a |$T$| -proof of |$x$| '. Finitist consistency is addressed by generalizing a result of Artemov: Partial finitism. If |$T$| is consistent, there is a primitive recursive function |$f$| such that, for all |$n\in \mathbb {N}$| , |$f(n)$| is the code of an |$\textsf {I}\varSigma _{1}$| -proof of |$\neg\, \textit{Prf}(\ulcorner \perp \urcorner ,\overline {n})$|. These results are not in conflict with Gödel's Incompleteness Theorems. Rather, they allow to extend their usual interpretation and show a deep connection to reflections in Hilbert's last papers of 1931. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: 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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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1093/logcom/exad021 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 20 StartPage: 1179 Subjects: – SubjectFull: Incompleteness theorems Type: general – SubjectFull: Recursive functions Type: general – SubjectFull: Number systems Type: general – SubjectFull: Axioms Type: general Titles: – TitleFull: A new perspective on completeness and finitist consistency. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Santos, Paulo Guilherme – PersonEntity: Name: NameFull: Sieg, Wilfried – PersonEntity: Name: NameFull: Kahle, Reinhard IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 09 Text: Sep2024 Type: published Y: 2024 Identifiers: – Type: issn-print Value: 0955792X Numbering: – Type: volume Value: 34 – Type: issue Value: 6 Titles: – TitleFull: Journal of Logic & Computation Type: main |
| ResultId | 1 |
