Editing Gödel's incompleteness theorems (section)

== References ==
=== Citations ===
{{Reflist}}

=== Articles by Gödel ===
* Kurt Gödel, 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I", [[Monatshefte für Mathematik und Physik]], v. 38 n. 1, pp.&nbsp;173&ndash;198. {{doi|10.1007/BF01700692}}
* —, 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I", in [[Solomon Feferman]], ed., 1986. ''Kurt Gödel Collected works, Vol. I''.  Oxford University Press, pp.&nbsp;144&ndash;195. {{ISBN|978-0195147209}}. The original German with a facing English translation, preceded by an introductory note by [[Stephen Cole Kleene]].
* —, 1951, "Some basic theorems on the foundations of mathematics and their implications", in [[Solomon Feferman]], ed., 1995. ''Kurt Gödel Collected works, Vol. III'', Oxford University Press, pp.&nbsp;304&ndash;323. {{ISBN|978-0195147223}}.

=== Translations, during his lifetime, of Gödel's paper into English ===
None of the following agree in all translated words and in typography. The typography is a serious matter, because Gödel expressly wished to emphasize "those metamathematical notions that had been defined in their usual sense before . . ." {{harv|van Heijenoort|1967|p=595}}. Three translations exist. Of the first John Dawson states that: "The Meltzer translation was seriously deficient and received a devastating review in the ''Journal of Symbolic Logic''; "Gödel also complained about Braithwaite's commentary {{harv|Dawson|1997|p=216}}. "Fortunately, the Meltzer translation was soon supplanted by a better one prepared by Elliott Mendelson for Martin Davis's anthology ''The Undecidable'' . . . he found the translation "not quite so good" as he had expected . . . [but because of time constraints he] agreed to its publication" (ibid). (In a footnote Dawson states that "he would regret his compliance, for the published volume was marred throughout by sloppy typography and numerous misprints" (ibid)). Dawson states that "The translation that Gödel favored was that by Jean van Heijenoort" (ibid). For the serious student another version exists as a set of lecture notes recorded by Stephen Kleene and J. B. Rosser "during lectures given by Gödel at to the Institute for Advanced Study during the spring of 1934" (cf commentary by {{harvnb|Davis|1965|p=39}} and beginning on p.&nbsp;41); this version is titled "On Undecidable Propositions of Formal Mathematical Systems". In their order of publication:
* B. Meltzer (translation) and [[R. B. Braithwaite]] (Introduction), 1962. ''On Formally Undecidable Propositions of Principia Mathematica and Related Systems'', Dover Publications, New York (Dover edition 1992), {{ISBN|0-486-66980-7}} (pbk.) This contains a useful translation of Gödel's German abbreviations on pp.&nbsp;33–34. As noted above, typography, translation and commentary is suspect. Unfortunately, this translation was reprinted with all its suspect content by
:* [[Stephen Hawking]] editor, 2005. ''God Created the Integers: The Mathematical Breakthroughs That Changed History'', Running Press, Philadelphia, {{ISBN|0-7624-1922-9}}. Gödel's paper appears starting on p. 1097, with Hawking's commentary starting on p. 1089.
* [[Martin Davis (mathematician)|Martin Davis]] editor, 1965. ''The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable problems and Computable Functions'', Raven Press, New York, no ISBN. Gödel's paper begins on page 5, preceded by one page of commentary.
* [[Jean van Heijenoort]] editor, 1967, 3rd edition 1967. ''From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931'', Harvard University Press, Cambridge Mass., {{ISBN|0-674-32449-8}} (pbk). van Heijenoort did the translation. He states that "Professor Gödel approved the translation, which in many places was accommodated to his wishes." (p.&nbsp;595). Gödel's paper begins on p.&nbsp;595; van Heijenoort's commentary begins on p.&nbsp;592.
* Martin Davis editor, 1965, ibid. "On Undecidable Propositions of Formal Mathematical Systems." A copy with Gödel's corrections of errata and Gödel's added notes begins on page 41, preceded by two pages of Davis's commentary. Until Davis included this in his volume this lecture existed only as mimeographed notes.

=== Articles by others ===
* {{cite journal |last=Boolos |first=George|author-link=George Boolos|title=A New Proof of the Gödel Incompleteness Theorem|year=1989 |journal=[[Notices of the American Mathematical Society]]|volume=36|pages=388–390, 676|quote=reprinted in {{harvtxt|Boolos|1998|pages=383–388}}}}
* {{cite book |last=Boolos |first=George|author-link=George Boolos|title=Logic, logic, and logic|year=1998|publisher=[[Harvard University Press]]|isbn=0-674-53766-1|pages=443}}
* Bernd Buldt, 2014, "[http://opus.ipfw.edu/cgi/viewcontent.cgi?article=1297&context=philos_facpubs The Scope of Gödel's First Incompleteness Theorem] {{Webarchive|url=https://web.archive.org/web/20160306110140/http://opus.ipfw.edu/cgi/viewcontent.cgi?article=1297&context=philos_facpubs |date=2016-03-06 }}", ''[[Logica Universalis]]'', v. 8, pp.&nbsp;499&ndash;552. {{doi|10.1007/s11787-014-0107-3}}
* {{cite journal |last1=Charlesworth |first1=Arthur |title=A Proof of Godel's Theorem in Terms of Computer Programs |journal=Mathematics Magazine |date=1981 |volume=54 |issue=3 |pages=109–121 |doi=10.2307/2689794 |jstor=2689794}}
* {{cite book |last1=Davis |first1=Martin |title=The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions |date=1965 |publisher=Raven Press |isbn=978-0-911216-01-1 |url=https://books.google.com/books?id=euBQAAAAMAAJ |language=en}}
* {{cite journal |author-link=Martin Davis (mathematician) |first=Martin |last=Davis |date=2006 |url=https://www.ams.org/notices/200604/fea-davis.pdf |title=The Incompleteness Theorem |journal=Notices of the AMS |volume=53 |issue=4 |page=414}}
* {{cite journal|url=http://www.digizeitschriften.de/dms/resolveppn/?PPN=GDZPPN002369001|title=Formale Beweise und die Entscheidbarkeit|journal=Mathematische Zeitschrift |year=1926 |volume=25 |pages=676–682 |doi=10.1007/bf01283861 |last1=Finsler|first1=Paul|s2cid=121054124}}
* {{cite book |editor-last=Grattan-Guinness |editor-first=Ivor |editor-link=Ivor Grattan-Guinness |title=Landmark Writings in Western Mathematics 1640-1940 |publisher=Elsevier |isbn=9780444508713 |date=2005}}
* {{cite book |author-link=Jean van Heijenoort |first=Jean |last=van Heijenoort |date=1967 |chapter=Gödel's Theorem |editor=Edwards, Paul |title=[[The Encyclopedia of Philosophy|Encyclopedia of Philosophy]] |volume=3 |publisher=Macmillan |pages=348–357}}
* {{cite journal |author-link=Geoffrey Hellman |first=Geoffrey |last=Hellman |date=1981 |title=How to Gödel a Frege-Russell: Gödel's Incompleteness Theorems and Logicism |journal=Noûs |volume=15 |issue=4 - Special Issue on Philosophy of Mathematics |pages=451–468 |doi=10.2307/2214847 |jstor=2214847 |issn=0029-4624}}
* [[David Hilbert]], 1900, "[http://aleph0.clarku.edu/~djoyce/hilbert/problems.html#prob2 Mathematical Problems.]" English translation of a lecture delivered before the International Congress of Mathematicians at Paris, containing Hilbert's statement of his Second Problem.
* Martin Hirzel, 2000, "[https://web.archive.org/web/20040916041216/http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf On formally undecidable propositions of Principia Mathematica and related systems I.]." An English translation of Gödel's paper. Archived from [http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf the original]. Sept. 16, 2004.
* {{cite journal |last1=Kikuchi |first1=Makoto |last2=Tanaka |first2=Kazuyuki |title=On Formalization of Model-Theoretic Proofs of Gödel's Theorems |journal=Notre Dame Journal of Formal Logic |date=July 1994 |volume=35 |issue=3 |pages=403–412 |doi=10.1305/ndjfl/1040511346 |mr=1326122|doi-access=free }}
* {{cite journal |last1=Kleene |first1=S. C. |author-link1=Stephen Cole Kleene |title=Recursive predicates and quantifiers |journal=Transactions of the American Mathematical Society |date=1943 |volume=53 |issue=1 |pages=41–73 |doi=10.1090/S0002-9947-1943-0007371-8}} Reprinted in {{harvnb|Davis|1965|pp=255–287}}
* {{cite web |first=Panu |last=Raatikainen |date=2020 |url=https://plato.stanford.edu/entries/goedel-incompleteness/ |title=Gödel's Incompleteness Theorems |publisher=Stanford Encyclopedia of Philosophy |access-date=November 7, 2022}}
* {{cite journal |first=Panu |last=Raatikainen |date=2005 |url=https://www.cairn.info/revue-internationale-de-philosophie-2005-4-page-513.htm |title=On the philosophical relevance of Gödel's incompleteness theorems |journal=Revue Internationale de Philosophie |volume=59 |issue=4 |pages=513–534|doi=10.3917/rip.234.0513 |s2cid=52083793 }}
* [[John Barkley Rosser]], 1936, "Extensions of some theorems of Gödel and Church", reprinted from the ''Journal of Symbolic Logic'', v. 1 (1936) pp.&nbsp;87–91, in Martin Davis 1965, ''The Undecidable'' (loc. cit.) pp.&nbsp;230–235.
* —, 1939, "An Informal Exposition of proofs of Gödel's Theorem and Church's Theorem", Reprinted from the ''Journal of Symbolic Logic'', v. 4 (1939) pp.&nbsp;53–60, in Martin Davis 1965, ''The Undecidable'' (loc. cit.) pp.&nbsp;223–230
* {{cite book |last=Smoryński |first=C. |chapter=The incompleteness theorems |editor=[[Jon Barwise]] |title=Handbook of mathematical logic |date=1977 |publisher=North-Holland Pub. Co |location=Amsterdam |isbn=978-0-444-86388-1 |pages=821–866}}
* {{cite journal |last=Willard |first=Dan E.  |year=2001 |title=Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles |journal=Journal of Symbolic Logic |volume=66 |number=2 |pages=536–596 |doi=10.2307/2695030 |jstor=2695030}}
* {{cite journal | last=Zach | first=Richard |author-link=Richard Zach | year=2003 | title=The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program | journal=[[Synthese]] | publisher=Springer Science and Business Media LLC | volume=137 | issue=1 |issn=0039-7857 | doi=10.1023/a:1026247421383 | pages=211–259 |url=http://people.ucalgary.ca/~rzach/static/conprf.pdf| arxiv=math/0102189 | s2cid=16657040 }}
* {{cite book | last=Zach | first=Richard |author-link=Richard Zach | year=2005 | chapter=Kurt Gödel, paper on the incompleteness theorems (1931) |pages=917&ndash;925 |editor-last=Grattan-Guinness |editor-first=Ivor |editor-link=Ivor Grattan-Guinness | title=Landmark Writings in Western Mathematics 1640-1940| publisher=Elsevier | doi=10.1016/b978-044450871-3/50152-2| isbn=9780444508713 | url=https://philarchive.org/rec/ZACKGP }}

=== Books about the theorems ===
* Francesco Berto. ''There's Something about Gödel:  The Complete Guide to the Incompleteness Theorem'' John Wiley and Sons. 2010.
* Norbert Domeisen, 1990. [https://archive.today/20240527193320/https://www.webcitation.org/6gQ72rSwF?url=http://www.textarchiv.homepage.bluewin.ch/Antinomien/Logik_der_Antinomien.html ''Logik der Antinomien'']. Bern: Peter Lang. 142 S. 1990. {{ISBN|3-261-04214-1}}. {{Zbl|0724.03003}}.
* {{cite book |last1=Franzén |first1=Torkel |author-link=Torkel Franzén |title=Gödel's theorem : an incomplete guide to its use and abuse |date=2005 |publisher=A K Peters |location=Wellesley, MA |isbn=1-56881-238-8 |mr=2146326}}
* [[Douglas Hofstadter]], 1979. ''[[Gödel, Escher, Bach|Gödel, Escher, Bach: An Eternal Golden Braid]]''. Vintage Books. {{ISBN|0-465-02685-0}}. 1999 reprint: {{ISBN|0-465-02656-7}}. {{MR|530196}}
*&nbsp;—, 2007. ''[[I Am a Strange Loop]]''. Basic Books. {{ISBN|978-0-465-03078-1}}. {{ISBN|0-465-03078-5}}. {{MR|2360307}}
* [[Stanley Jaki]], OSB, 2005. ''The drama of the quantities''. [http://www.realviewbooks.com/ Real View Books.]
* [[Per Lindström]], 1997. ''[https://projecteuclid.org/euclid.lnl/1235416274 Aspects of Incompleteness]'', Lecture Notes in Logic v. 10.
* [[J.R. Lucas]], FBA, 1970. ''The Freedom of the Will''. Clarendon Press, Oxford, 1970.
* [[Adrian William Moore]], 2022. ''Gödel´s Theorem: A Very Short Introduction''. Oxford University Press, Oxford, 2022.
* [[Ernest Nagel]], [[James R. Newman|James Roy Newman]], Douglas Hofstadter, 2002 (1958). ''Gödel's Proof'', revised ed.  {{ISBN|0-8147-5816-9}}. {{MR|1871678}}
* [[Rudy Rucker]], 1995 (1982). ''Infinity and the Mind: The Science and Philosophy of the Infinite''. Princeton Univ. Press. {{MR|658492}}
* {{cite book |last1=Smith |first1=Peter |title=An introduction to Gödel's Theorems |date=2007 |url=http://www.godelbook.net/ |publisher=Cambridge University Press |location=Cambridge, U.K. |isbn=978-0-521-67453-9 |mr=2384958 |access-date=2005-10-29 |archive-date=2005-10-23 |archive-url=https://web.archive.org/web/20051023200804/http://www.godelbook.net/ |url-status=dead }}
* {{cite book |last1=Shankar |first1=N. |title=Metamathematics, machines, and Gödel's proof |date=1994 |volume=38 |series=Cambridge tracts in theoretical computer science |publisher=Cambridge University Press |location=Cambridge |isbn=0-521-58533-3}}
* [[Raymond Smullyan]], 1987. ''Forever Undecided'' {{ISBN|0192801414}} - puzzles based on undecidability in formal systems
* —, 1992. ''Godel's Incompleteness Theorems''. Oxford Univ. Press. {{ISBN|0195046722}}
* —, 1994. ''Diagonalization and Self-Reference''. Oxford Univ. Press. {{MR|1318913}}. {{ISBN|0198534507}}
* —, 2013. [https://books.google.com/books?id=xUapAAAAQBAJ ''The Godelian Puzzle Book: Puzzles, Paradoxes and Proofs'']. Courier Corporation. {{ISBN|978-0-486-49705-1}}.
* {{cite book |author-link=Hao Wang (academic) |first=Hao |last=Wang |date=1996 |title=A Logical Journey: From Gödel to Philosophy |publisher=MIT Press |isbn=0-262-23189-1}} {{MR|1433803}}

=== Miscellaneous references ===
* {{cite journal |first=Francesco |last=Berto |date=2009 |title=The Gödel Paradox and Wittgenstein's Reasons |journal=[[Philosophia Mathematica]] |volume=III |issue=17}}
* {{cite book | last=Dawson | first=John W. Jr. | title=Logical dilemmas: The life and work of Kurt Gödel | publisher=Taylor & Francis | year=1996 | isbn=978-1-56881-025-6}}
* {{cite book | last=Dawson | first=John W. Jr. | title=Logical dilemmas: The life and work of Kurt Gödel | publisher=[[A. K. Peters]] | publication-place=Wellesley, Massachusetts | year=1997 | isbn=978-1-56881-256-4 | oclc=36104240}}
* [[Rebecca Goldstein]], 2005, ''Incompleteness: the Proof and Paradox of Kurt Gödel'', W. W. Norton & Company. {{ISBN|0-393-05169-2}}
* {{cite journal | last1=Floyd | first1=Juliet | last2=Putnam | first2=Hilary | title=A Note on Wittgenstein's "Notorious Paragraph" about the Godel Theorem | journal=[[The Journal of Philosophy]] | publisher=JSTOR | volume=97 | issue=11 | year=2000 | issn=0022-362X | doi=10.2307/2678455 | pages=624–632| jstor=2678455 }}
* {{cite book |last1=Harrison |first1=J. |title=Handbook of practical logic and automated reasoning |date=2009 |publisher=Cambridge University Press |location=Cambridge |isbn=978-0521899574}}
* [[David Hilbert]] and [[Paul Bernays]], ''[[Grundlagen der Mathematik]]'', Springer-Verlag.
* {{cite book |last1=Hopcroft |first1=John E. |author-link1=John Hopcroft |last2=Ullman |first2=Jeffrey |author-link2=Jeffrey Ullman |title=[[Introduction to Automata Theory, Languages, and Computation]] |date=1979 |publisher=Addison-Wesley |location=Reading, Mass. |isbn=0-201-02988-X}}
* {{cite book |last = Hofstadter |first = Douglas R. |author-link = Douglas Hofstadter |title = I Am a Strange Loop |title-link = I Am a Strange Loop |chapter = Chapter 12. On Downward Causality |chapter-url = https://publicism.info/philosophy/strange/14.html |year = 2007 |publisher = Basic Books |orig-year = 2003 |isbn = 978-0-465-03078-1 |access-date = 2018-10-24 |archive-date = 2019-05-08 |archive-url = https://web.archive.org/web/20190508182834/https://publicism.info/philosophy/strange/14.html |url-status = dead }}
* {{cite journal |first=James P. |last=Jones |url=https://www.ams.org/bull/1980-03-02/S0273-0979-1980-14832-6/S0273-0979-1980-14832-6.pdf |title=Undecidable Diophantine Equations |journal=Bulletin of the American Mathematical Society |volume=3 |issue=2 |date=1980 |pages=859–862|doi=10.1090/S0273-0979-1980-14832-6 }}
* {{cite journal |author-link=Stephen Cole Kleene |first=Stephen Cole |last=Kleene |date=1967 |title=Mathematical Logic|journal=Nature |volume=216 |issue=5111 |page=201 |doi=10.1038/216201b0 |bibcode=1967Natur.216..201G }} Reprinted by Dover, 2002. {{ISBN|0-486-42533-9}}
* {{cite book |first=Russell |last=O'Connor |title=Theorem Proving in Higher Order Logics |chapter=Essential Incompleteness of Arithmetic Verified by Coq |series=Lecture Notes in Computer Science |date=2005 |volume=3603 |pages=245–260|doi=10.1007/11541868_16 |arxiv=cs/0505034 |isbn=978-3-540-28372-0 |s2cid=15610367 }}
* {{cite journal |author-link=Lawrence Paulson |first=Lawrence |last=Paulson |date=2014 |title=A machine-assisted proof of Gödel's incompleteness theorems for the theory of hereditarily finite sets |journal=Review of Symbolic Logic |volume=7 |issue=3 |pages=484–498 |doi=10.1017/S1755020314000112|s2cid=13913592 |url=https://www.repository.cam.ac.uk/handle/1810/245422 |arxiv=2104.14260 }}
* {{cite journal |last=Priest |first=Graham |author-link=Graham Priest |date=1984 |title=Logic of Paradox Revisited |journal=Journal of Philosophical Logic |volume=13 |number=2  |pages=153–179|doi=10.1007/BF00453020 }}
* {{cite book |last=Priest |first=Graham |date=2004 |chapter=Wittgenstein's Remarks on Gödel's Theorem |editor=Max Kölbel |title=Wittgenstein's lasting significance |publisher=Psychology Press |pages=207–227 |isbn=978-1-134-40617-3}}
* {{cite book |author-link=Graham Priest |first=Graham |last=Priest |date=2006 |title=In Contradiction: A Study of the Transconsistent |publisher=Oxford University Press |isbn=0-19-926329-9}}
* {{cite book |last=Putnam |first=Hilary |author-link=Hilary Putnam |date=1960 |chapter=Minds and Machines |editor=Sidney Hook |editor-link=Sidney Hook |title=Dimensions of Mind: A Symposium |publisher=New York University Press}} Reprinted in Anderson, A. R., ed., 1964. ''Minds and Machines''. Prentice-Hall: 77.
* [[Wolfgang Rautenberg]], 2010, ''[https://link.springer.com/book/10.1007%2F978-1-4419-1221-3 A Concise Introduction to Mathematical Logic]'', 3rd. ed., Springer, {{ISBN|978-1-4419-1220-6}}
* {{cite journal |last1=Rodych |first1=Victor |title=Misunderstanding Gödel: New Arguments about Wittgenstein and New Remarks by Wittgenstein |journal=Dialectica |date=2003 |volume=57 |issue=3 |pages=279–313|doi=10.1111/j.1746-8361.2003.tb00272.x }} {{doi|10.1111/j.1746-8361.2003.tb00272.x}}
* {{cite journal|mr=0357114|last=Shelah| first=Saharon|authorlink = Saharon Shelah|title=Infinite Abelian groups, Whitehead problem and some constructions |journal=[[Israel Journal of Mathematics]] |volume=18 |year=1974| issue=3|pages=243–256|doi=10.1007/BF02757281|doi-access=free}}
* {{cite journal |last=Shapiro |first=Stewart |author-link=Stewart Shapiro |date=2002 |title=Incompleteness and Inconsistency |journal=Mind |volume=111 |issue=444 |pages=817–32 |doi=10.1093/mind/111.444.817 }}
* {{cite book |author-link1=Alan Sokal |first1=Alan |last1=Sokal |author-link2=Jean Bricmont |first2=Jean |last2=Bricmont |date=1999 |title=[[Fashionable Nonsense]]: Postmodern Intellectuals' Abuse of Science |publisher=Picador |isbn=0-312-20407-8}}
* {{cite book |last1=Shoenfield |first1=Joseph R. |author-link1=Joseph R. Shoenfield |title=Mathematical logic |date=1967 |publication-date=2001 |publisher=Association for Symbolic Logic |location=Natick, Mass. |isbn=978-1-56881-135-2}}
* {{cite book |author-link1=Jeremy Stangroom |first1=Jeremy |last1=Stangroom |author-link2=Ophelia Benson |first2=Ophelia |last2=Benson |title=Why Truth Matters |publisher=Continuum |date=2006 |isbn=0-8264-9528-1}}
* George Tourlakis, ''Lectures in Logic and Set Theory, Volume 1, Mathematical Logic'', Cambridge University Press, 2003. {{ISBN|978-0-521-75373-9}}
* {{cite book |last=Wigderson |first=Avi |author-link=Avi Wigderson |date=2010 |chapter-url=http://www.math.ias.edu/~avi/BOOKS/Godel_Widgerson_Text.pdf |chapter=The Gödel Phenomena in Mathematics: A Modern View |title=Kurt Gödel and the Foundations of Mathematics: Horizons of Truth |publisher=Cambridge University Press}}
* [[Hao Wang (academic)|Hao Wang]], 1996, ''A Logical Journey: From Gödel to Philosophy'', The MIT Press, Cambridge MA, {{ISBN|0-262-23189-1}}.
* {{cite book | last=Zach | first=Richard  | year=2007| chapter=Hilbert's Program Then and Now|pages=411–447 | doi=10.1016/b978-044451541-4/50014-2 | editor-last=Jacquette | editor-first=Dale | title=Philosophy of logic |series=Handbook of the Philosophy of Science |volume=5 | publisher=Elsevier | arxiv=math/0508572 | publication-place=Amsterdam | isbn=978-0-444-51541-4 | s2cid=291599 | oclc=162131413}}