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...
Gespeichert in:
| Veröffentlicht in: | Journal of automated reasoning 2015-06, Vol.55 (1), p.1-37 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 37 |
|---|---|
| container_issue | 1 |
| container_start_page | 1 |
| container_title | Journal of automated reasoning |
| container_volume | 55 |
| creator | Paulson, Lawrence C. |
| description | 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. |
| doi_str_mv | 10.1007/s10817-015-9322-8 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_1680662361</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>3683385421</sourcerecordid><originalsourceid>FETCH-LOGICAL-c386t-e5513e62fe161f08c94b960d607e6768c6a50863a5ece6d7babc7a05ac9d17143</originalsourceid><addsrcrecordid>eNp1kEFOwzAQRS0EEqVwAHaWWBvGcWM7y6qCUqlQFu3acpxJmyqJi90u2HENLsIFuAknIVVYsEEaaTb_fc08Qq453HIAdRc5aK4Y8JRlIkmYPiEDnirBQCo4JQPgUjM1EuKcXMS4BQDBIRuQxZg-odvYtopY0JfgfUm7mX59Flh_v39EOmudb3Y17rHFGOlygz5gE-kqVu2aPvumam1NZ9HmWNd4Sc5KW0e8-t1Dsnq4X04e2XwxnU3Gc-aElnuGacoFyqRELnkJ2mWjPJNQSFAoldRO2hS0FDZFh7JQuc2dspBalxVc8ZEYkpu-dxf86wHj3mz9IXSXRNN9ClImQvIuxfuUCz7GgKXZhaqx4c1wMEdvpvdmOm_m6M3ojkl6JnbZdo3hT_O_0A9qmnD0</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1680662361</pqid></control><display><type>article</type><title>A Mechanised Proof of Gödel’s Incompleteness Theorems Using Nominal Isabelle</title><source>SpringerLink Journals - AutoHoldings</source><creator>Paulson, Lawrence C.</creator><creatorcontrib>Paulson, Lawrence C.</creatorcontrib><description>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.</description><identifier>ISSN: 0168-7433</identifier><identifier>EISSN: 1573-0670</identifier><identifier>DOI: 10.1007/s10817-015-9322-8</identifier><language>eng</language><publisher>Dordrecht: Springer Netherlands</publisher><subject>Artificial Intelligence ; Computer Science ; Mathematical Logic and Formal Languages ; Mathematical Logic and Foundations ; Symbolic and Algebraic Manipulation</subject><ispartof>Journal of automated reasoning, 2015-06, Vol.55 (1), p.1-37</ispartof><rights>Springer Science+Business Media Dordrecht 2015</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c386t-e5513e62fe161f08c94b960d607e6768c6a50863a5ece6d7babc7a05ac9d17143</citedby><cites>FETCH-LOGICAL-c386t-e5513e62fe161f08c94b960d607e6768c6a50863a5ece6d7babc7a05ac9d17143</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10817-015-9322-8$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10817-015-9322-8$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Paulson, Lawrence C.</creatorcontrib><title>A Mechanised Proof of Gödel’s Incompleteness Theorems Using Nominal Isabelle</title><title>Journal of automated reasoning</title><addtitle>J Autom Reasoning</addtitle><description>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.</description><subject>Artificial Intelligence</subject><subject>Computer Science</subject><subject>Mathematical Logic and Formal Languages</subject><subject>Mathematical Logic and Foundations</subject><subject>Symbolic and Algebraic Manipulation</subject><issn>0168-7433</issn><issn>1573-0670</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2015</creationdate><recordtype>article</recordtype><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><recordid>eNp1kEFOwzAQRS0EEqVwAHaWWBvGcWM7y6qCUqlQFu3acpxJmyqJi90u2HENLsIFuAknIVVYsEEaaTb_fc08Qq453HIAdRc5aK4Y8JRlIkmYPiEDnirBQCo4JQPgUjM1EuKcXMS4BQDBIRuQxZg-odvYtopY0JfgfUm7mX59Flh_v39EOmudb3Y17rHFGOlygz5gE-kqVu2aPvumam1NZ9HmWNd4Sc5KW0e8-t1Dsnq4X04e2XwxnU3Gc-aElnuGacoFyqRELnkJ2mWjPJNQSFAoldRO2hS0FDZFh7JQuc2dspBalxVc8ZEYkpu-dxf86wHj3mz9IXSXRNN9ClImQvIuxfuUCz7GgKXZhaqx4c1wMEdvpvdmOm_m6M3ojkl6JnbZdo3hT_O_0A9qmnD0</recordid><startdate>20150601</startdate><enddate>20150601</enddate><creator>Paulson, Lawrence C.</creator><general>Springer Netherlands</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>GNUQQ</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K7-</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M7S</scope><scope>P5Z</scope><scope>P62</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope></search><sort><creationdate>20150601</creationdate><title>A Mechanised Proof of Gödel’s Incompleteness Theorems Using Nominal Isabelle</title><author>Paulson, Lawrence C.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c386t-e5513e62fe161f08c94b960d607e6768c6a50863a5ece6d7babc7a05ac9d17143</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2015</creationdate><topic>Artificial Intelligence</topic><topic>Computer Science</topic><topic>Mathematical Logic and Formal Languages</topic><topic>Mathematical Logic and Foundations</topic><topic>Symbolic and Algebraic Manipulation</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Paulson, Lawrence C.</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies & Aerospace Collection</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>ProQuest Central Student</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>Computer Science Database</collection><collection>ProQuest Engineering Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Engineering Database</collection><collection>Advanced Technologies & Aerospace Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection><jtitle>Journal of automated reasoning</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Paulson, Lawrence C.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A Mechanised Proof of Gödel’s Incompleteness Theorems Using Nominal Isabelle</atitle><jtitle>Journal of automated reasoning</jtitle><stitle>J Autom Reasoning</stitle><date>2015-06-01</date><risdate>2015</risdate><volume>55</volume><issue>1</issue><spage>1</spage><epage>37</epage><pages>1-37</pages><issn>0168-7433</issn><eissn>1573-0670</eissn><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.</abstract><cop>Dordrecht</cop><pub>Springer Netherlands</pub><doi>10.1007/s10817-015-9322-8</doi><tpages>37</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0168-7433 |
| ispartof | Journal of automated reasoning, 2015-06, Vol.55 (1), p.1-37 |
| issn | 0168-7433 1573-0670 |
| language | eng |
| recordid | cdi_proquest_journals_1680662361 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Artificial Intelligence Computer Science Mathematical Logic and Formal Languages Mathematical Logic and Foundations Symbolic and Algebraic Manipulation |
| title | A Mechanised Proof of Gödel’s Incompleteness Theorems Using Nominal Isabelle |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-04T18%3A20%3A11IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=A%20Mechanised%20Proof%20of%20G%C3%B6del%E2%80%99s%20Incompleteness%20Theorems%20Using%20Nominal%20Isabelle&rft.jtitle=Journal%20of%20automated%20reasoning&rft.au=Paulson,%20Lawrence%20C.&rft.date=2015-06-01&rft.volume=55&rft.issue=1&rft.spage=1&rft.epage=37&rft.pages=1-37&rft.issn=0168-7433&rft.eissn=1573-0670&rft_id=info:doi/10.1007/s10817-015-9322-8&rft_dat=%3Cproquest_cross%3E3683385421%3C/proquest_cross%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=1680662361&rft_id=info:pmid/&rfr_iscdi=true |