Editing Mathematical logic (section)

==References==
{{Reflist|15em}}

=== Undergraduate texts ===
* {{Cite book |last=Walicki |first=Michał |date=2011 |title=Introduction to Mathematical Logic |location=[[Singapore]] |publisher=[[World Scientific Publishing]] |isbn=9789814343879
}}
* {{Cite book |last1=Boolos |first1=George |author1-link=George Boolos |last2=Burgess |first2=John |last3=Jeffrey |first3=Richard |author3-link=Richard Jeffrey |date=2002 |title=Computability and Logic |edition=4th |publisher=[[Cambridge University Press]] |isbn=9780521007580
}}
* {{Cite book |last1=Crossley |first1=J.N. |last2=Ash |first2=C.J. |last3=Brickhill |first3=C.J. |last4=Stillwell |first4=J.C. |last5=Williams |first5=N.H. |date=1972 |title=What is mathematical logic? |location=London, Oxford, New York City |publisher=[[Oxford University Press]] |isbn=9780198880875 |zbl=0251.02001
}}
* {{Cite book |last=Enderton |first=Herbert |date=2001 |title=A mathematical introduction to logic |edition=2nd |location=[[Boston]] MA |publisher=[[Academic Press]] |isbn=978-0-12-238452-3 
}}
* {{Cite book |last=Fisher |first=Alec |date=1982 |title=Formal Number Theory and Computability: A Workbook |edition=1st |others=(suitable as a first course for independent study) |publisher=Oxford University Press |isbn=978-0-19-853188-3 
}}
* {{Cite book |last=Hamilton |first=A.G. |date=1988 |title=Logic for Mathematicians |edition=2nd |publisher=Cambridge University Press |isbn=978-0-521-36865-0 
}}
*{{Cite book |last1=Ebbinghaus |first1=H.-D. |last2=Flum |first2=J. |last3=Thomas |first3=W. |date=1994 |title=Mathematical Logic |edition=2nd |location=[[New York City]] |publisher=[[Springer Science+Business Media|Springer]] |isbn=9780387942582 |url=https://www.springer.com/mathematics/book/978-0-387-94258-2 
}}
* {{Cite book |last1=Katz |first1=Robert |date=1964 |title=Axiomatic Analysis |location=[[Boston]] MA |publisher=[[D. C. Heath and Company]]
}}
* {{Cite book |last=Mendelson |first=Elliott |author-link=Elliott Mendelson |date=1997 |title=Introduction to Mathematical Logic |edition=4th |location=London |publisher=[[Chapman & Hall]] |isbn=978-0-412-80830-2 
}}
* {{Cite book |last=Rautenberg |first=Wolfgang |author-link=Wolfgang Rautenberg |date=2010 |title=A Concise Introduction to Mathematical Logic |edition=3rd |location=[[New York City]] |publisher=[[Springer Science+Business Media|Springer]] |doi=10.1007/978-1-4419-1221-3 |isbn=9781441912206
}}
* {{Cite book |last=Schwichtenberg |first=Helmut |author-link=Helmut Schwichtenberg |year=2003–2004 |title=Mathematical Logic |location=[[Munich]] |publisher=Mathematisches Institut der Universität München |access-date=2016-02-24 |url=http://www.mathematik.uni-muenchen.de/~schwicht/lectures/logic/ws03/ml.pdf 
}}
* Shawn Hedman, ''A first course in logic: an introduction to model theory, proof theory, computability, and complexity'', [[Oxford University Press]], 2004, {{isbn|0-19-852981-3}}. Covers logics in close relation with [[computability theory]] and [[Computational complexity theory|complexity theory]]
* {{Cite book |last=van Dalen |first=Dirk |date=2013 |title=Logic and Structure |series=Universitext |location=Berlin |publisher=[[Springer Science+Business Media|Springer]] |doi=10.1007/978-1-4471-4558-5 |isbn=978-1-4471-4557-8 }}

=== Graduate texts ===
* {{Cite book |last=Hinman |first=Peter G. |date=2005 |title= Fundamentals of mathematical logic |publisher=A K Peters, Ltd. |isbn=1-56881-262-0
}}
* {{Cite book |last=Andrews |first=Peter B. |date=2002 |title=An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof |edition=2nd |location=[[Boston]] |publisher=Kluwer Academic Publishers |isbn=978-1-4020-0763-7 
}}
* {{Cite book| editor-last=Barwise |editor-first=Jon |editor-link=Jon Barwise |date=1989 |title=Handbook of Mathematical Logic |location=[[Amsterdam]] |publisher=[[Elsevier]] |series=Studies in Logic and the Foundations of Mathematics |isbn=9780444863881 
}}
* {{Cite book |last=Hodges |first=Wilfrid |author-link=Wilfrid Hodges |date=1997 |title=A shorter model theory |publisher=[[Cambridge University Press]] |isbn=9780521587136
}}
* {{Cite book |last=Jech |first=Thomas |author-link=Thomas Jech |date=2003 |title=Set Theory: Millennium Edition |series=Springer Monographs in Mathematics |location=Berlin, New York |publisher=[[Springer Science+Business Media|Springer]] |isbn=9783540440857 
}}
*[[Stephen Cole Kleene|Kleene, Stephen Cole]].(1952), ''[https://books.google.com/books?id=HZAjPwAACAAJ&source=gbs_ViewAPI Introduction to Metamathematics.]'' New York: Van Nostrand. (Ishi Press: 2009 reprint).
*[[Stephen Cole Kleene|Kleene, Stephen Cole]]. (1967),  ''[http://worldcat.org/oclc/523472 Mathematical Logic.]'' John Wiley. Dover reprint, 2002. {{isbn|0-486-42533-9}}.
* {{Cite book |last=Shoenfield |first=Joseph R. |author-link=Joseph R. Shoenfield |date=2001 |orig-date=1967 |title=Mathematical Logic |edition=2nd |publisher=[[A K Peters]] |isbn=9781568811352 
}}
* {{Cite book |last1=Troelstra |first1=Anne Sjerp |author1-link=A. S. Troelstra |last2=Schwichtenberg |first2=Helmut |date=2000 |title=Basic Proof Theory |edition=2nd |series=Cambridge Tracts in Theoretical Computer Science |publisher=Cambridge University Press |isbn=978-0-521-77911-1 }}

=== Research papers, monographs, texts, and surveys ===
* {{cite book|url=http://www.collegepublications.co.uk/logic/mlf/?00029|title=Logical consequences. Theory and applications: An introduction|first1=Luis M.|last1= Augusto|publisher=College Publications|location=London|date=2017|isbn=978-1-84890-236-7}}
* {{cite book |last=Boehner |first=Philotheus |author-link=Philotheus Boehner |date=1950 |title=Medieval Logic |location=Manchester
}}
* {{Cite book |last=Cohen |first=Paul J. |author-link=Paul Cohen |date=1966 |title=Set Theory and the Continuum Hypothesis |location=[[Menlo Park, California|Menlo Park CA]] |publisher=W. A. Benjamin
}}
* {{cite book |last=Cohen |first=Paul J. |author-link=Paul Cohen |date=2008 |orig-date=1966 |title=Set theory and the continuum hypothesis |location=[[Mineola, New York|Mineola NY]] |publisher=Dover Publications |isbn=9780486469218
}}
*J.D. Sneed, ''The Logical Structure of Mathematical Physics''. Reidel, Dordrecht, 1971 (revised edition 1979).
* {{Cite journal |last=Davis |first=Martin |author-link=Martin Davis (mathematician) |date=1973 |title=[[Hilbert's tenth problem]] is unsolvable |journal=[[American Mathematical Monthly|The American Mathematical Monthly]] |volume=80 |issue=3 |pages=233–269 |doi=10.2307/2318447 |jstor=2318447 }}<br> Reprinted as an appendix in {{cite book |author=Martin Davis |date=1985 |title=Computability and Unsolvability |publisher=Dover |isbn=9780486614717
}}
*{{Cite journal |last=Felscher |first=Walter |date=2000 |title=Bolzano, Cauchy, Epsilon, Delta |journal=The American Mathematical Monthly |volume=107 |issue=9 |pages=844–862 |doi=10.2307/2695743 |jstor=2695743
}}
*{{Cite journal |last=Ferreirós |first=José |date=2001 |title=The Road to Modern Logic-An Interpretation |journal=Bulletin of Symbolic Logic |volume=7 |issue=4 |pages=441–484 |doi=10.2307/2687794 |jstor=2687794 |hdl=11441/38373 |s2cid=43258676 |url=https://idus.us.es/xmlui/bitstream/11441/38373/1/The%20road%20to%20modern%20logic.pdf
}}
* {{Cite journal |last1=Hamkins |first1=Joel David |last2=Löwe |first2=Benedikt |date=2007 |title=The modal logic of forcing |journal=Transactions of the American Mathematical Society |volume=360 |issue=4 |pages=1793–1818 |doi=10.1090/s0002-9947-07-04297-3 |s2cid = 14724471 |arxiv=math/0509616
}}
* {{Cite book |last1=Katz |first1=Victor J. |date=1998 |title=A History of Mathematics |publisher=Addison–Wesley |isbn=9780321016188
}}
* {{Cite journal |last=Morley |first=Michael |author-link=Michael D. Morley |date=1965 |title=Categoricity in Power |journal=[[Transactions of the American Mathematical Society]] |volume=114 |issue=2 |pages=514–538 |doi=10.2307/1994188 |jstor=1994188 |doi-access=free
}}
* {{Cite journal |last=Soare |first=Robert I. |date=1996 |title=Computability and recursion |journal=Bulletin of Symbolic Logic |volume=2 |issue = 3 |pages=284–321 |doi=10.2307/420992 |jstor=420992 |citeseerx=10.1.1.35.5803
|s2cid=5894394 }}
*{{Cite journal |last=Solovay |first=Robert M. |author-link=Robert M. Solovay |date=1976 |title=Provability Interpretations of Modal Logic |journal=[[Israel Journal of Mathematics]] |volume=25 |issue=3–4 |pages=287–304 |doi=10.1007/BF02757006 |doi-access= |s2cid= 121226261
}}
*{{Cite journal |last=Woodin |first=W. Hugh |author-link=W. Hugh Woodin |date=2001 |title=The Continuum Hypothesis, Part I |journal=Notices of the American Mathematical Society |volume=48 |issue=6 |url=https://www.ams.org/notices/200106/fea-woodin.pdf}}

=== Classical papers, texts, and collections ===
* {{cite journal |last1=Banach |first1=Stefan |author-link1=Stefan Banach |last2=Tarski |first2=Alfred |author-link2=Alfred Tarski |date=1924 |title=Sur la décomposition des ensembles de points en parties respectivement congruentes |journal=[[Fundamenta Mathematicae]] |volume=6 |pages=244–277 |doi=10.4064/fm-6-1-244-277 |language=fr |doi-access=free |url=http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm6127.pdf
}}
{{cite book |editor-last=Bochenski |editor-first=Jozef Maria |date=1959 |translator=Otto Bird |title=A Precis of Mathematical Logic |series=Synthese Library, Vol. 1 |location=[[Dordrecht]] |publisher=[[Springer Science+Business Media|Springer]] |doi=10.1007/978-94-017-0592-9 |isbn=9789048183296 
}}
*{{Cite book |last=Burali-Forti |first=Cesare |date=1897 |title=A question on transfinite numbers }} Reprinted in {{harvnb|van Heijenoort|1976|pp=104–111
}}
{{Cite journal |last=Cantor |first=Georg |year=1874 |title=Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen |journal=[[Journal für die Reine und Angewandte Mathematik]] |volume=1874 |issue=77 |pages=258–262 |doi=10.1515/crll.1874.77.258 |s2cid=199545885 |url=http://gdz.sub.uni-goettingen.de/download/PPN243919689_0077/PPN243919689_0077___LOG_0014.pdf 
}}
{{cite book |last=Carroll |first=Lewis |author-link=Lewis Carroll |date=1896 |title=Symbolic Logic |publisher=Kessinger Legacy Reprints |isbn=9781163444955 |url=https://www.gutenberg.org/ebooks/28696
}}
*{{cite book |last=Dedekind |first=Richard |author-link=Richard Dedekind |date=1872 |title=Stetigkeit und irrationale Zahlen |language=de  }} English translation as: "Consistency and irrational numbers".
*{{cite book |last=Dedekind |first=Richard |author-link=Richard Dedekind |date=1888 |title=Was sind und was sollen die Zahlen? }} Two English translations:
**1963 (1901). ''Essays on the Theory of Numbers''. Beman, W. W., ed. and trans. Dover.
**1996. In ''From Kant to Hilbert: A Source Book in the Foundations of Mathematics'', 2 vols, Ewald, William B., ed., [[Oxford University Press]]: 787–832.
*{{cite book |last1=Fraenkel |first1=Abraham A. |author1-link=Abraham Fraenkel |date=1922 |contribution=Der Begriff 'definit' und die Unabhängigkeit des Auswahlsaxioms |title=Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse |pages=253–257 |language=de }} Reprinted in English translation as "The notion of 'definite' and the independence of the axiom of choice" in {{harvnb|van Heijenoort|1976|pp=284–289}}.
* [[Gottlob Frege|Frege, Gottlob]] (1879), ''[[Begriffsschrift]], eine der arithmetischen nachgebildete Formelsprache des reinen Denkens''. Halle a. S.: Louis Nebert. Translation: ''Concept Script, a formal language of pure thought modelled upon that of arithmetic'', by S. Bauer-Mengelberg in {{harvnb|van Heijenoort|1976}}.
* [[Gottlob Frege|Frege, Gottlob]] (1884), ''Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl''. Breslau: W. Koebner. Translation: [[J. L. Austin]], 1974. ''The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number'', 2nd ed. Blackwell.
* {{cite journal |last=Gentzen |first=Gerhard |author-link=Gerhard Gentzen |date=1936 |title=Die Widerspruchsfreiheit der reinen Zahlentheorie |journal=Mathematische Annalen |volume=112 |pages=132–213 |doi=10.1007/BF01565428 |s2cid=122719892  }} Reprinted in English translation in Gentzen's ''Collected works'', M. E. Szabo, ed., North-Holland, Amsterdam, 1969.
*{{Cite book |last=Gödel |first=Kurt |author-link=Kurt Gödel |date=1929 |title=Über die Vollständigkeit des Logikkalküls |trans-title=Completeness of the logical calculus |series=doctoral dissertation |publisher=University Of Vienna
}}
* {{Cite journal |last=Gödel |first=Kurt |author-link=Kurt Gödel |date=1930 |title=Die Vollständigkeit der Axiome des logischen Funktionen-kalküls |trans-title=The completeness of the axioms of the calculus of logical functions |journal= Monatshefte für Mathematik und Physik |volume=37 |pages=349–360 |doi=10.1007/BF01696781 |s2cid=123343522 |language=de
}}
* {{Cite journal |last=Gödel |first=Kurt |author-link=Kurt Gödel |date=1931 |title=Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I |trans-title=[[On Formally Undecidable Propositions of Principia Mathematica and Related Systems]] |journal=Monatshefte für Mathematik und Physik |volume=38 |issue=1 |pages=173–198 |doi=10.1007/BF01700692 |s2cid=197663120 |language=de }}
*{{Cite journal |last=Gödel |first=Kurt |author-link=Kurt Gödel |date=1958 |title=Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes |journal=Dialectica |volume=12 |issue=3–4 |pages=280–287 |doi=10.1111/j.1746-8361.1958.tb01464.x |doi-access=free |language=de }} Reprinted in English translation in Gödel's ''Collected Works'', vol II, [[Solomon Feferman]] et al., eds. Oxford University Press, 1993.
* {{cite book |editor-last=van Heijenoort |editor-first=Jean |editor-link=Jean van Heijenoort |date=1976 |orig-date=1967 |title=From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 |edition=3rd |location=[[Cambridge, Massachusetts|Cambridge MA]] |publisher=[[Harvard University Press]] |isbn=9780674324497 |id=(pbk.) 
}}
*{{cite book |last=Hilbert |first=David |author-link=David Hilbert |date=1899 |title=Grundlagen der Geometrie |location=[[Leipzig]] |publisher=Teubner |language=de}} English 1902 edition (''The Foundations of Geometry'') republished 1980, Open Court, Chicago.
*{{Cite journal |last=Hilbert |first=David |author-link=David Hilbert |date=1929 |title=Probleme der Grundlegung der Mathematik |journal=Mathematische Annalen |volume=102 |pages=1–9 |doi=10.1007/BF01782335 |s2cid=122870563 |url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002273500&L=1 }} Lecture given at the International Congress of Mathematicians, 3 September 1928. Published in English translation as "The Grounding of Elementary Number Theory", in Mancosu 1998, pp.&nbsp;266–273.
* {{Cite book |last1=Hilbert |first1=David |author1-link=David Hilbert |last2=Bernays |first2=Paul |author2-link=Paul Bernays |date=1934 |title=Grundlagen der Mathematik. I |series=Die Grundlehren der mathematischen Wissenschaften |volume=40 |location=Berlin, New York City |publisher=[[Springer Science+Business Media|Springer]] |isbn=9783540041344 |mr=0237246 |jfm=60.0017.02
}}
* {{Cite journal |last=Kleene |first=Stephen Cole |author-link=Stephen Kleene |date=1943 |title=Recursive Predicates and Quantifiers |journal= Transactions of the American Mathematical Society|volume=53 |issue=1 |pages=41–73 |doi=10.2307/1990131 |jstor=1990131 |doi-access=free
}}
* {{Cite book |last=Lobachevsky |first=Nikolai |author-link=Nikolai Lobachevsky |date=1840 |title=Geometrishe Untersuchungen zur Theorie der Parellellinien |language=de }} Reprinted in English translation as {{cite book |date=1955 |chapter=Geometric Investigations on the Theory of Parallel Lines |editor=Robert Bonola |title=Non-Euclidean Geometry |publisher=Dover |isbn=0-486-60027-0
}}
* {{Cite journal |last1=Löwenheim |first1=Leopold |author1-link=Leopold Löwenheim |date=1915 |title=Über Möglichkeiten im Relativkalkül |journal=Mathematische Annalen |volume=76 |issue=4 |pages=447–470 |doi=10.1007/BF01458217 |issn=0025-5831 |s2cid=116581304 |language=de |url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002266121&L=1 }} Translated as "On possibilities in the calculus of relatives" in {{cite book |author=Jean van Heijenoort |author-link=Jean van Heijenoort |date=1967 |title=A Source Book in Mathematical Logic, 1879–1931 |publisher=Harvard Univ. Press |pages=228–251
}}
*{{Cite book |editor-last=Mancosu |editor-first=Paolo |editor-link=Paolo Mancosu |date=1998 |title=From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s |publisher=Oxford University Press
}}
*{{cite book |last=Pasch |first=Moritz |author-link=Moritz Pasch |date=1882 |title=Vorlesungen über neuere Geometrie
}}
*{{cite book |last=Peano |first=Giuseppe |author-link=Giuseppe Peano |date=1889 |title=Arithmetices principia, nova methodo exposita |title-link=Arithmetices principia, nova methodo exposita |language=lt }} Excerpt reprinted in English translation as "The principles of arithmetic, presented by a new method"in {{harvnb|van Heijenoort|1976|pp=83–97}}.
* {{Cite journal |last=Richard |first=Jules |date=1905 |title=Les principes des mathématiques et le problème des ensembles |journal=Revue Générale des Sciences Pures et Appliquées |volume=16 |page=541 |language=fr}} Reprinted in English translation as "The principles of mathematics and the problems of sets" in {{harvnb|van Heijenoort|1976|pp=142–144}}.
* {{Cite journal |last1=Skolem |first1=Thoralf |author1-link=Thoralf Skolem |date=1920 |title=Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen |journal=Videnskapsselskapet Skrifter, I. Matematisk-naturvidenskabelig Klasse |volume=6 |pages=1–36 |language=de
}}
{{cite web |last=Soare |first=Robert Irving |date=22 December 2011 |title=Computability Theory and Applications: The Art of Classical Computability |website=Department of Mathematics |publisher=University of Chicago |access-date=23 August 2017 |url=http://www.people.cs.uchicago.edu/~soare/Turing/frontice.pdf
}}
{{cite book |last=Swineshead |first=Richard |author-link=Richard Swineshead |date=1498 |title=Calculationes Suiseth Anglici |publisher=Papie: Per Franciscum Gyrardengum |language=lt 
}}
* {{Cite book |last=Tarski |first=Alfred |author-link=Alfred Tarski |date=1948 |title=A decision method for elementary algebra and geometry |location=[[Santa Monica, California|Santa Monica CA]] |publisher=[[RAND Corporation]]
}}
*{{Cite journal |last1=Turing |first1=Alan M. |author1-link=A. M. Turing |date=1939 |title=Systems of Logic Based on Ordinals |journal=[[Proceedings of the London Mathematical Society]] |volume=45 |issue=2 |pages=161–228 |doi=10.1112/plms/s2-45.1.161 |hdl=21.11116/0000-0001-91CE-3 |hdl-access=free
}}
* {{cite book |last=Weyl |first=Hermann |date=1918 |title=Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis |location=Leipzig |language=de
}}
* {{Cite journal |last=Zermelo |first=Ernst |author-link=Ernst Zermelo |date=1904 |title=Beweis, daß jede Menge wohlgeordnet werden kann |journal=Mathematische Annalen |volume=59 |issue=4 |pages=514–516 |doi=10.1007/BF01445300 |s2cid=124189935 |language=de |url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002260018&L=1 }} Reprinted in English translation as "Proof that every set can be well-ordered" in {{harvnb|van Heijenoort|1976|pp=139–141}}.
* {{Cite journal |last=Zermelo |first=Ernst |author-link=Ernst Zermelo |date=1908a |title=Neuer Beweis für die Möglichkeit einer Wohlordnung |journal=[[Mathematische Annalen]] |volume=65 |pages=107–128 |doi=10.1007/BF01450054 |issn=0025-5831 |s2cid=119924143 |language=de |url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002261952&L=1 }} Reprinted in English translation as "A new proof of the possibility of a well-ordering" in {{harvnb|van Heijenoort|1976|pp=183–198}}.
* {{Cite journal |last=Zermelo |first=Ernst |author-link=Ernst Zermelo |date=1908b |title=Untersuchungen über die Grundlagen der Mengenlehre |journal=Mathematische Annalen |volume=65 |issue=2 |pages=261–281 |doi=10.1007/BF01449999 |s2cid=120085563 |url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0065&DMDID=DMDLOG_0018&L=1 }}