A new perspective on completeness and finitist consistency.
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| Title: | A new perspective on completeness and finitist consistency. |
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| 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})$|. These results are not in conflict with Gödel's Incompleteness Theorems. Rather, they allow to extend their usual interpretation and show a deep connection to reflections in Hilbert's last papers of 1931. [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
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| Abstract: | In this paper, we study the metamathematics of consistent arithmetical theories |$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] |
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| ISSN: | 0955792X |
| DOI: | 10.1093/logcom/exad021 |
