A Mechanised Proof of Gödel’s Incompleteness Theorems Using Nominal Isabelle
An Isabelle/HOL formalisation of Gödel’s two incompleteness theorems is presented. The work follows Świerczkowski’s detailed proof of the theorems using hereditarily finite (HF) set theory (Dissertationes Mathematicae 422 , 1–58, 2003 ). Avoiding the usual arithmetical encodings of syntax eliminates...
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| Vydáno v: | Journal of automated reasoning Ročník 55; číslo 1; s. 1 - 37 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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01.06.2015
Springer Nature B.V |
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| ISSN: | 0168-7433, 1573-0670 |
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| Abstract | An Isabelle/HOL formalisation of Gödel’s two incompleteness theorems is presented. The work follows Świerczkowski’s detailed proof of the theorems using hereditarily finite (HF) set theory (Dissertationes Mathematicae
422
, 1–58,
2003
). Avoiding the usual arithmetical encodings of syntax eliminates the necessity to formalise elementary number theory within an embedded logical calculus. The Isabelle formalisation uses two separate treatments of variable binding: the nominal package (Logical Methods in Computer Science
8
(2:14), 1–35,
2012
) is shown to scale to a development of this complexity, while de Bruijn indices (Indagationes Mathematicae
34
, 381–392,
1972
) turn out to be ideal for coding syntax. Critical details of the Isabelle proof are described, in particular gaps and errors found in the literature. |
|---|---|
| AbstractList | An Isabelle/HOL formalisation of Gödel's two incompleteness theorems is presented. The work follows wierczkowski's detailed proof of the theorems using hereditarily finite (HF) set theory (Dissertationes Mathematicae 422, 1-58, 2003 ). Avoiding the usual arithmetical encodings of syntax eliminates the necessity to formalise elementary number theory within an embedded logical calculus. The Isabelle formalisation uses two separate treatments of variable binding: the nominal package (Logical Methods in Computer Science 8(2:14), 1-35, 2012 ) is shown to scale to a development of this complexity, while de Bruijn indices (Indagationes Mathematicae 34, 381-392, 1972 ) turn out to be ideal for coding syntax. Critical details of the Isabelle proof are described, in particular gaps and errors found in the literature. An Isabelle/HOL formalisation of Gödel’s two incompleteness theorems is presented. The work follows Świerczkowski’s detailed proof of the theorems using hereditarily finite (HF) set theory (Dissertationes Mathematicae 422 , 1–58, 2003 ). Avoiding the usual arithmetical encodings of syntax eliminates the necessity to formalise elementary number theory within an embedded logical calculus. The Isabelle formalisation uses two separate treatments of variable binding: the nominal package (Logical Methods in Computer Science 8 (2:14), 1–35, 2012 ) is shown to scale to a development of this complexity, while de Bruijn indices (Indagationes Mathematicae 34 , 381–392, 1972 ) turn out to be ideal for coding syntax. Critical details of the Isabelle proof are described, in particular gaps and errors found in the literature. |
| Author | Paulson, Lawrence C. |
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| Cites_doi | 10.1112/S1461157000000449 10.1016/1385-7258(72)90034-0 10.1002/malq.200610026 10.1017/S1755020314000112 10.4064/dm422-0-1 10.1007/BF00881873 10.1007/s001650200016 10.1017/CBO9780511576430 10.1007/978-3-540-74591-4_16 10.1201/b10700 10.1007/11541868 10.1017/CBO9780511569883 10.1023/A:1006496715975 10.1007/11814771_17 10.1007/978-94-010-1191-4 10.1017/CBO9781139084673 10.1007/3-540-45949-9 10.1007/978-3-642-03359-9_22 10.1007/11541868_25 10.1017/CBO9780511625183 10.1007/s10817-008-9097-2 |
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| Keywords | Isabelle/HOL Nominal syntax Gödel’s incompleteness theorems Formalisation of mathematics |
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| References | Paulson (CR25) 2003; 6 Paulson (CR27) 2014; 7 CR19 CR18 CR17 Urban, Kaliszyk (CR34) 2012; 8 Gabbay, Pitts (CR6) 2001; 13 CR16 CR14 CR13 CR12 CR11 CR33 CR10 CR31 CR30 Świerczkowski (CR32) 2003; 422 CR2 de Bruijn (CR3) 1972; 34 Lawrence (CR24) 1993; 11 CR4 CR5 CR8 CR7 CR29 CR28 CR9 CR26 CR23 CR22 CR21 CR20 Kirby (CR15) 2007; 53 Bagaria (CR1) 2003; 22 9322_CR2 9322_CR26 9322_CR28 C Urban (9322_CR34) 2012; 8 9322_CR29 9322_CR22 9322_CR23 J Bagaria (9322_CR1) 2003; 22 9322_CR20 9322_CR21 L Kirby (9322_CR15) 2007; 53 NG Bruijn de (9322_CR3) 1972; 34 9322_CR19 9322_CR16 LC Paulson (9322_CR25) 2003; 6 9322_CR17 9322_CR18 C Lawrence (9322_CR24) 1993; 11 9322_CR11 9322_CR33 9322_CR12 9322_CR13 9322_CR14 9322_CR30 9322_CR31 9322_CR10 S Świerczkowski (9322_CR32) 2003; 422 MJ Gabbay (9322_CR6) 2001; 13 9322_CR9 LC Paulson (9322_CR27) 2014; 7 9322_CR8 9322_CR7 9322_CR5 9322_CR4 |
| References_xml | – ident: CR22 – volume: 6 start-page: 198 year: 2003 end-page: 248 ident: CR25 article-title: The relative consistency of the axiom of choice — mechanized using Isabelle/ZF publication-title: LMS J. Comput. Math. doi: 10.1112/S1461157000000449 – ident: CR18 – volume: 22 start-page: 5 issue: 3 year: 2003 end-page: 15 ident: CR1 article-title: A short guide to Gödel’s second incompleteness theorem publication-title: Teorema – ident: CR4 – ident: CR14 – ident: CR2 – ident: CR16 – ident: CR12 – ident: CR30 – volume: 34 start-page: 381 year: 1972 end-page: 392 ident: CR3 article-title: Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser Theorem publication-title: Indag. Math. doi: 10.1016/1385-7258(72)90034-0 – volume: 53 start-page: 52 issue: 1 year: 2007 end-page: 65 ident: CR15 article-title: Addition and multiplication of sets publication-title: Math. Logic Q. doi: 10.1002/malq.200610026 – ident: CR10 – ident: CR33 – ident: CR29 – ident: CR8 – volume: 7 start-page: 484 issue: 3 year: 2014 end-page: 498 ident: CR27 article-title: A machine-assisted proof of Gödel’s incompleteness theorems for the theory of hereditarily finite sets publication-title: Rev. Symb. Log. doi: 10.1017/S1755020314000112 – volume: 8 start-page: 1 issue: 2:14 year: 2012 end-page: 35 ident: CR34 article-title: General bindings and alpha-equivalence in Nominal Isabelle publication-title: Log. Methods Comput. Sci. – ident: CR23 – ident: CR21 – volume: 422 start-page: 1 year: 2003 end-page: 58 ident: CR32 article-title: Finite sets and Godel’s̈ incompleteness theorems publication-title: Dissertationes Mathematicae doi: 10.4064/dm422-0-1 – ident: CR19 – ident: CR17 – ident: CR31 – ident: CR13 – volume: 11 start-page: 353 issue: 3 year: 1993 end-page: 389 ident: CR24 article-title: Paulson. Set theory for verification: I. From foundations to functions publication-title: J. Autom. Reas. doi: 10.1007/BF00881873 – ident: CR11 – volume: 13 start-page: 341 year: 2001 end-page: 363 ident: CR6 article-title: A new approach to abstract syntax with variable binding publication-title: Form. Asp. Comput. doi: 10.1007/s001650200016 – ident: CR9 – ident: CR5 – ident: CR7 – ident: CR28 – ident: CR26 – ident: CR20 – volume: 53 start-page: 52 issue: 1 year: 2007 ident: 9322_CR15 publication-title: Math. Logic Q. doi: 10.1002/malq.200610026 – ident: 9322_CR12 doi: 10.1017/CBO9780511576430 – ident: 9322_CR21 doi: 10.1007/978-3-540-74591-4_16 – volume: 7 start-page: 484 issue: 3 year: 2014 ident: 9322_CR27 publication-title: Rev. Symb. Log. doi: 10.1017/S1755020314000112 – ident: 9322_CR5 doi: 10.1201/b10700 – ident: 9322_CR14 doi: 10.1007/11541868 – ident: 9322_CR26 – ident: 9322_CR23 – volume: 22 start-page: 5 issue: 3 year: 2003 ident: 9322_CR1 publication-title: Teorema – ident: 9322_CR7 – ident: 9322_CR30 doi: 10.1017/CBO9780511569883 – ident: 9322_CR10 – ident: 9322_CR16 – ident: 9322_CR18 doi: 10.1023/A:1006496715975 – ident: 9322_CR31 – volume: 8 start-page: 1 issue: 2:14 year: 2012 ident: 9322_CR34 publication-title: Log. Methods Comput. Sci. – ident: 9322_CR11 doi: 10.1007/11814771_17 – ident: 9322_CR9 doi: 10.1007/978-94-010-1191-4 – ident: 9322_CR28 doi: 10.1017/CBO9781139084673 – ident: 9322_CR20 doi: 10.1007/3-540-45949-9 – volume: 6 start-page: 198 year: 2003 ident: 9322_CR25 publication-title: LMS J. Comput. Math. doi: 10.1112/S1461157000000449 – ident: 9322_CR29 – ident: 9322_CR17 doi: 10.1007/978-3-642-03359-9_22 – ident: 9322_CR22 – volume: 13 start-page: 341 year: 2001 ident: 9322_CR6 publication-title: Form. Asp. Comput. doi: 10.1007/s001650200016 – ident: 9322_CR19 doi: 10.1007/11541868_25 – volume: 422 start-page: 1 year: 2003 ident: 9322_CR32 publication-title: Dissertationes Mathematicae doi: 10.4064/dm422-0-1 – volume: 11 start-page: 353 issue: 3 year: 1993 ident: 9322_CR24 publication-title: J. Autom. Reas. doi: 10.1007/BF00881873 – volume: 34 start-page: 381 year: 1972 ident: 9322_CR3 publication-title: Indag. Math. doi: 10.1016/1385-7258(72)90034-0 – ident: 9322_CR8 – ident: 9322_CR2 doi: 10.1017/CBO9780511625183 – ident: 9322_CR4 – ident: 9322_CR13 – ident: 9322_CR33 doi: 10.1007/s10817-008-9097-2 |
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| Snippet | An Isabelle/HOL formalisation of Gödel’s two incompleteness theorems is presented. The work follows Świerczkowski’s detailed proof of the theorems using... An Isabelle/HOL formalisation of Gödel's two incompleteness theorems is presented. The work follows wierczkowski's detailed proof of the theorems using... |
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