Editing Peano axioms (section)

=== Sources ===
{{refbegin}}
* {{cite book
|last= Davis
|first= Martin
|author-link= Martin Davis (mathematician)
|year= 1974
|title= Computability. Notes by Barry Jacobs.
|publisher= [[Courant Institute of Mathematical Sciences]], [[New York University]]
}}

* {{cite book
|last= Dedekind
|first= Richard
|author-link= Richard Dedekind
|year= 1888
|title= Was sind und was sollen die Zahlen?
|trans-title= What are and what should the numbers be?
|publisher= Vieweg
|url= http://digisrv-1.biblio.etc.tu-bs.de:8080/docportal/servlets/MCRFileNodeServlet/DocPortal_derivate_00005731/V.C.125.pdf
|access-date= 4 July 2016
|archive-date= 20 October 2016
|archive-url= https://web.archive.org/web/20161020135148/http://digisrv-1.biblio.etc.tu-bs.de:8080/docportal/servlets/MCRFileNodeServlet/DocPortal_derivate_00005731/V.C.125.pdf
|url-status= dead
}}
** Two English translations:
*** {{cite book
|last= Beman
|first= Wooster, Woodruff
|year= 1901
|title= Essays on the Theory of Numbers
|publisher= [[Dover Publications]]
|url= http://www.gutenberg.org/files/21016/21016-pdf.pdf
}}
*** {{cite book
|last= Ewald
|first= William B.
|year= 1996
|title= From Kant to Hilbert: A Source Book in the Foundations of Mathematics
|url= https://books.google.com/books?id=Nt_uAAAAMAAJ
|publisher= [[Oxford University Press]]
|pages= 787–832
|isbn= 978-0-19-853271-2
}}

* {{cite book
|last= Fritz
|first= Charles A. Jr.
|year= 1952
|title= Bertrand Russell's construction of the external world
|publisher= New York, Humanities Press
|url= https://archive.org/details/bertrandrussells0000frit
|url-access= registration
}}

* {{cite journal
|last= Gentzen
|first= Gerhard
|author-link= Gerhard Gentzen
|year= 1936
|title= Die Widerspruchsfreiheit der reinen Zahlentheorie
|journal= [[Mathematische Annalen]]
|volume= 112
|pages= 132–213
|doi= 10.1007/bf01565428
|others= Reprinted in English translation in his 1969 ''Collected works'', M. E. Szabo, ed.
|s2cid= 122719892
}}

* {{cite journal
|last= Gödel
|first= Kurt
|author-link= Kurt Gödel
|year= 1931
|title= Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I
|journal= Monatshefte für Mathematik
|volume= 38
|pages= 173–198
|url= http://www.w-k-essler.de/pdfs/goedel.pdf
|doi= 10.1007/bf01700692
|others= See [[On Formally Undecidable Propositions of Principia Mathematica and Related Systems]] for details on English translations.
|s2cid= 197663120
|access-date= 2013-10-31
|archive-url= https://web.archive.org/web/20180411113347/http://www.w-k-essler.de/pdfs/goedel.pdf
|archive-date= 2018-04-11
}}

* {{cite journal
|last= Gödel
|first= Kurt
|year= 1958
|author-link= Kurt Gödel
|title= Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes
|journal= [[Dialectica]]
|volume= 12
|pages= 280–287
|others= Reprinted in English translation in 1990. Gödel's ''Collected Works'', Vol II. [[Solomon Feferman]] et al., eds.
|issue= 3–4
|publisher= [[Oxford University Press]]
|doi= 10.1111/j.1746-8361.1958.tb01464.x
|doi-access= free
}}

* {{cite book
|last= Grassmann
|first= Hermann Günther
|author-link= Hermann Grassmann
|year= 1861
|title= Lehrbuch der Arithmetik für höhere Lehranstalten
|publisher= Verlag von Theod. Chr. Fr. Enslin
|url= https://books.google.com/books?id=jdQ2AAAAMAAJ
}}

* {{cite book
|last= Gray
|first= Jeremy
|author-link= Jeremy Gray
|year= 2013
|title= Henri Poincaré: A scientific biography
|chapter= The Essayist
|chapter-url= https://books.google.com/books?id=w2Tya9gOKqEC&pg=PA133
|page= 133
|publisher= [[Princeton University Press]]
|isbn= 978-0-691-15271-4
}}

* {{cite book
|last= Harsanyi
|first= John C.
|author-link= John C. Harsanyi
|year= 1983
|title= Methodology, Epistemology, and Philosophy of Science
|chapter= Mathematics, the Empirical Facts, and Logical Necessity
|editor1-last= Hempel
|editor1-first= Carl G.
|editor2-last= Putnam
|editor2-first= Hilary
|editor3-last= Essler
|editor3-first= Wilhelm K.
|pages= 167–192
|doi= 10.1007/978-94-015-7676-5_8
|isbn= 978-90-481-8389-0
|s2cid= 121297669
}}

* {{cite book
|last= Hatcher
|first= William S.
|date= 2014
|orig-date= 1982
|title= The Logical Foundations of Mathematics
|url= https://books.google.com/books?id=wf3iBQAAQBAJ&pg=PP1
|publisher= Elsevier
|isbn= 978-1-4831-8963-5
}} Derives the Peano axioms (called '''S''') from several [[axiomatic set theories]] and from [[category theory]].

* {{cite book
|last= Hermes
|first= Hans
|year= 1973
|issn= 1431-4657
|isbn= 3-540-05819-2
|title= Introduction to Mathematical Logic
|publisher= Springer
|series= Hochschultext
}}

* {{cite journal
|last= Hilbert
|first= David
|year= 1902
|title= Mathematische Probleme
|trans-title= Mathematical Problems
|journal= Bulletin of the American Mathematical Society
|volume= 8
|issue= 10
|pages= 437–479
|url= https://www.ams.org/journals/bull/1902-08-10/S0002-9904-1902-00923-3/home.html
|translator-first= Maby
|translator-last= Winton
|doi= 10.1090/s0002-9904-1902-00923-3
|doi-access= free
}}

* {{cite book
|last= Kaye
|first= Richard
|year= 1991
|title= Models of Peano arithmetic
|publisher= [[Oxford University Press]]
|isbn= 0-19-853213-X
}}

* {{cite book
|last= Landau
|first= Edmund
|author-link= Edmund Landau
|year= 1965
|title= Grundlagen Der Analysis
|publisher= [[American Mathematical Society|AMS Chelsea Publishing]]
|others= Derives the basic number systems from the Peano axioms. English/German vocabulary included.
|isbn= 978-0-8284-0141-8
}}

* {{cite book
|last= Mendelson
|first= Elliott
|author-link= Elliott Mendelson
|date= December 1997
|orig-date=December 1979
|title= Introduction to Mathematical Logic (Discrete Mathematics and Its Applications)
|edition= 4th
|publisher= Springer
|isbn= 978-0-412-80830-2
}}

* {{cite book
|last1= Meseguer
|first1= José
|last2= Goguen
|first2= Joseph A.
|date= Dec 1986
|url= https://courses.engr.illinois.edu/cs576/sp2017/readings/background/initiality-induction-computability.pdf
|contribution= Initiality, induction, and computability
|pages= 459–541
|isbn= 978-0-521-26793-9
|editor= Maurice Nivat and John C. Reynolds
|title= Algebraic Methods in Semantics
|location= Cambridge
|publisher= Cambridge University Press
}}

* {{cite book
|last1= Partee
|first1= Barbara
|last2= Ter Meulen
|first2= Alice
|author2-link= Alice ter Meulen
|last3= Wall
|first3= Robert
|year= 2012
|title= Mathematical Methods in Linguistics
|publisher= Springer
|url= https://books.google.com/books?id=d5xrCQAAQBAJ
|isbn= 978-94-009-2213-6
}}

* {{cite book
|last= Peano
|first= Giuseppe
|author-link= Giuseppe Peano
|year= 1908
|title= Formulario Mathematico
|edition= V
|page= 27
|publisher= Turin, Bocca frères, Ch. Clausen
|url= https://archive.org/details/formulairedemat04peangoog
}}

* {{cite journal
|last= Peirce
|first= C.&nbsp;S.
|author-link= Charles Sanders Peirce
|year= 1881
|title= On the Logic of Number
|url= https://archive.org/details/jstor-2369151
|journal= American Journal of Mathematics
|volume= 4
|issue= 1
|pages= 85–95
|doi= 10.2307/2369151
|mr= 1507856
|jstor= 2369151
}}

* {{cite book
|last= Shields
|first= Paul
|year= 1997
|title= Studies in the Logic of Charles Sanders Peirce
|url= https://archive.org/details/studiesinlogicof00nath
|url-access= registration
|chapter= 3. Peirce's Axiomatization of Arithmetic
|chapter-url= https://books.google.com/books?id=pWjOg-zbtMAC&pg=PA43
|editor1-last= Houser
|editor1-first= Nathan
|editor2-last= Roberts
|editor2-first= Don D.
|editor3-last= Van Evra
|editor3-first= James
|publisher= Indiana University Press
|isbn= 0-253-33020-3
|pages= 43–52
}}

* {{cite book
|last= Suppes
|first= Patrick
|author-link= Patrick Suppes
|year= 1960
|title= Axiomatic Set Theory
|publisher= [[Dover Publications]]
|isbn= 0-486-61630-4
|url-access= registration
|url= https://archive.org/details/axiomaticsettheo00supp_0
}} Derives the Peano axioms from [[ZFC]]

* {{cite book
|last1= Tarski
|first1= Alfred
|author-link1= Alfred Tarski
|last2= Givant
|first2= Steven
|year= 1987
|title= A Formalization of Set Theory without Variables
|series= AMS Colloquium Publications
|volume= 41
|publisher= [[American Mathematical Society]]
|isbn= 978-0-8218-1041-5
|url-access= registration
|url= https://archive.org/details/formalizationofs0000tars
}}

* {{cite book
|last= Van Heijenoort
|first= Jean
|author-link= Jean van Heijenoort
|year= 1967
|title= From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931
|publisher= Harvard University Press
|isbn= 978-0-674-32449-7
}}
** Contains translations of the following two papers, with valuable commentary:
*** {{cite book
|last= Dedekind
|first= Richard
|author-link= Richard Dedekind
|year= 1890
|title= Letter to Keferstein.
|pages= 98–103
|others= On p.&nbsp;100, he restates and defends his axioms of 1888.
}}
*** {{cite book
|last= Peano
|first= Giuseppe
|author-link= Giuseppe Peano
|year= 1889
|title= Arithmetices principia, nova methodo exposita
|trans-title= The principles of arithmetic, presented by a new method
|pages= 83–97
|url= https://archive.org/details/arithmeticespri00peangoog
|others= An excerpt of the treatise where Peano first presented his axioms, and recursively defined arithmetical operations.
|publisher= Fratres Bocca
}}

* {{cite web
|last= Van Oosten
|first= Jaap
|date= June 1999
|title= Introduction to Peano Arithmetic (Gödel Incompleteness and Nonstandard Models)
|url= https://webspace.science.uu.nl/~ooste110/syllabi/peanomoeder.pdf
|publisher= Utrecht University
|access-date= 2 September 2023
}}

* {{cite journal
|last= Wang
|first= Hao
|author-link= Hao Wang (academic)
|date= June 1957
|title= The Axiomatization of Arithmetic
|journal= The Journal of Symbolic Logic
|volume= 22
|issue= 2
|pages= 145–158
|doi= 10.2307/2964176
|publisher= [[Association for Symbolic Logic]]
|jstor= 2964176
|s2cid= 26896458
}}

*{{cite journal
|last= Willard
|first= Dan E.
|author-link= Dan Willard
|year= 2001
|doi= 10.2307/2695030
|issue= 2
|journal= The Journal of Symbolic Logic
|mr= 1833464
|pages= 536–596
|title= Self-verifying axiom systems, the incompleteness theorem and related reflection principles
|url= https://www.cs.albany.edu/~dew/m/jsl1.pdf
|volume= 66
|jstor= 2695030
|s2cid= 2822314
|archive-date= 2020-11-09
|access-date= 2018-06-30
|archive-url= https://web.archive.org/web/20201109033211/http://www.cs.albany.edu/~dew/m/jsl1.pdf
|url-status= dead
}}
{{refend}}