Natural number (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Natural number" in English language version.

Last modified: 14:09:02, 11 August 2026 (UTC)

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acta.hu (Global: low place; English: low place)

von Neumann (1923). von Neumann, John (1923). "Zur Einführung der transfiniten Zahlen" [On the Introduction of the Transfinite Numbers]. Acta Litterarum AC Scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio Scientiarum Mathematicarum. 1: 199–208.
- von Neumann, John (January 2002) [1923]. "On the introduction of transfinite numbers". In van Heijenoort, Jean (ed.). From Frege to Gödel: A source book in mathematical logic, 1879–1931 (3rd ed.). Harvard University Press. pp. 346–354. ISBN 978-0-674-32449-7. – English translation of von Neumann (1923).

ams.org (Global: 434^th place; English: 249^th place)

mathscinet.ams.org

Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4 (1): 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856.
Baratella, Stefano; Ferro, Ruggero (1993). "A theory of sets with the negation of the axiom of infinity". Mathematical Logic Quarterly. 39 (3): 338–352. doi:10.1002/malq.19930390138. MR 1270381.

archive.org (Global: 6^th place; English: 6^th place)

Hamilton (1988), pp. 117 ff calls them "Peano's Postulates" and begins with "1. 0 is a natural number."
Halmos (1966), p. 46 uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I) $0 \in ω$ (where, of course, $0 = \emptyset$ " ( $ω$ is the set of all natural numbers).
Morash (1991), Section 10.1: An Axiomatization for the System of Positive Integers gives "a two-part axiom" in which the natural numbers begin with 1. Hamilton, A. G. (1988). Logic for Mathematicians (Revised ed.). Cambridge University Press. ISBN 978-0-521-36865-0. Halmos, Paul (1966). Naive Set Theory. Princeton: Van Nostrand. LCCN 60-11059. Morash, Ronald P. (1991). Bridge to Abstract Mathematics: Mathematical proof and structures (2nd ed.). Mcgraw-Hill. ISBN 978-0-07-043043-3.
Russell (1920), p. 12: "In actual fact, it is simpler logically to find out whether two collections have the same number of terms than it is to define what that number is." Russell, Bertrand (1920) [1919]. Introduction to Mathematical Philosophy (2nd ed.). London: George Allen & Unwin.
Russell (1920), p. 14: "It is very easy to see that if (for example) a collection has three members, the class of all those collections that are similar to it will be the class of trios. And whatever number of terms a collection may have, those collections that are 'similar' to it will have the same number of terms. We may take this as a definition of 'having the same number of terms.'" Russell, Bertrand (1920) [1919]. Introduction to Mathematical Philosophy (2nd ed.). London: George Allen & Unwin.
Quine (1960), pp. 262–263, says the only thing required of an acceptable description of natural numbers is that they form a progression and so "any progression – i.e., any infinite series each of whose members has only finitely many precursors – will do nicely". Quine, Willard (1960). Word and Object. Technology Press (MIT). LCCN 60-9621.
Rudin, W. (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. p. 25. ISBN 978-0-07-054235-8.
Peano, Giuseppe (1901). Formulaire des mathematiques [Formulary for Mathematics] (in French). Paris: Gauthier-Villars. p. 39.
Russell (1920), p. 13. Russell, Bertrand (1920) [1919]. Introduction to Mathematical Philosophy (2nd ed.). London: George Allen & Unwin.
Russell (1920). Russell, Bertrand (1920) [1919]. Introduction to Mathematical Philosophy (2nd ed.). London: George Allen & Unwin.
Emerson, William (1763). The method of increments. p. 113.
Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4 (1): 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856.
Shields, Paul (1997). "3. Peirce's Axiomatization of Arithmetic". In Houser, Nathan; Roberts, Don D.; Van Evra, James (eds.). Studies in the Logic of Charles Sanders Peirce. Indiana University Press. pp. 43–52. ISBN 0-253-33020-3.
Was sind und was sollen die Zahlen? [What are numbers and what should they be?] (in German). F. Vieweg. 1893. pp. 71–73.
Arithmetices principia: nova methodo (in Latin). Fratres Bocca. 1889. p. 12.

bnf.fr (Global: 61^st place; English: 357^th place)

gallica.bnf.fr

Chuquet, Nicolas (1881) [1484]. Le Triparty en la science des nombres (in French).
Fontenelle, Bernard de (1727). Eléments de la géométrie de l'infini [Elements of geometry of infinity] (in French). p. 3.

books.google.com (Global: 3^rd place; English: 3^rd place)

Hamilton (1988), pp. 117 ff calls them "Peano's Postulates" and begins with "1. 0 is a natural number."
Halmos (1966), p. 46 uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I) $0 \in ω$ (where, of course, $0 = \emptyset$ " ( $ω$ is the set of all natural numbers).
Morash (1991), Section 10.1: An Axiomatization for the System of Positive Integers gives "a two-part axiom" in which the natural numbers begin with 1. Hamilton, A. G. (1988). Logic for Mathematicians (Revised ed.). Cambridge University Press. ISBN 978-0-521-36865-0. Halmos, Paul (1966). Naive Set Theory. Princeton: Van Nostrand. LCCN 60-11059. Morash, Ronald P. (1991). Bridge to Abstract Mathematics: Mathematical proof and structures (2nd ed.). Mcgraw-Hill. ISBN 978-0-07-043043-3.
Cooke, Heather (2000). Primary Mathematics. SAGE. p. 14. ISBN 978-1-84787-949-3.
Zegarelli, Mark (2014). Basic Math and Pre-Algebra For Dummies. Wiley. p. 21. ISBN 978-1-118-79199-8. Counting numbers (also called natural numbers): The set of numbers beginning 1, 2, 3, 4, ... and going on infinitely.
Mendelson (2008), p. x says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers."
Bluman (2010), p. 1: "Numbers make up the foundation of mathematics." Bluman, Allan (2010). Pre-Algebra DeMYSTiFieD (2nd ed.). McGraw-Hill Professional. ISBN 978-0-07-174251-1.
Ganssle, Jack G. & Barr, Michael (2003). "integer". Embedded Systems Dictionary. Taylor & Francis. pp. 138 (integer), 247 (signed integer), & 276 (unsigned integer). ISBN 978-1-57820-120-4. Archived from the original on 29 March 2017. Retrieved 28 March 2017 – via Google Books.
Rice, Harris (1922). "Errors in computations and the rounded number". The Mathematics Teacher. National Council of Teachers of Mathematics. p. 393. A counting number is the number given in answer to the question "How many?" In this class of numbers belongs zero and positive integers/
Stewart, Ian; Tall, David (12 March 2015). The Foundations of Mathematics. OUP Oxford. p. 160. ISBN 978-0-19-101648-6. Retrieved 30 July 2025.
Fokas, Athanassios; Kaxiras, Efthimios (12 December 2022). Modern Mathematical Methods For Scientists And Engineers: A Street-smart Introduction. World Scientific. p. 4. ISBN 978-1-80061-182-5. Retrieved 30 July 2025.
Fletcher, Harold; Howell, Arnold A. (9 May 2014). Mathematics with Understanding. Elsevier. p. 116. ISBN 978-1-4832-8079-0. ...the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication
Davisson, Schuyler Colfax (1910). College Algebra. Macmillian Company. p. 2. Addition of natural numbers is associative.
Brandon, Bertha (M.); Brown, Kenneth E.; Gundlach, Bernard H.; Cooke, Ralph J. (1962). Laidlaw mathematics series. Vol. 8. Laidlaw Bros. p. 25.
Gray, Jeremy (2008). Plato's Ghost: The modernist transformation of mathematics. Princeton University Press. p. 153. ISBN 978-1-4008-2904-0. Archived from the original on 29 March 2017 – via Google Books.
Shields, Paul (1997). "3. Peirce's Axiomatization of Arithmetic". In Houser, Nathan; Roberts, Don D.; Van Evra, James (eds.). Studies in the Logic of Charles Sanders Peirce. Indiana University Press. pp. 43–52. ISBN 0-253-33020-3.
Fine, Henry Burchard (1904). A College Algebra. Ginn. p. 6.
Advanced Algebra: A Study Guide to be Used with USAFI Course MC 166 Or CC166. United States Armed Forces Institute. 1958. p. 12.
Wohlgemuth, Andrew (10 June 2014). Introduction to Proof in Abstract Mathematics. Courier Corporation. p. 237. ISBN 978-0-486-14168-8.

digizeitschriften.de (Global: low place; English: low place)

Weber, Heinrich L. (1891–1892). "Kronecker". Jahresbericht der Deutschen Mathematiker-Vereinigung [Annual report of the German Mathematicians Association]. pp. 2:5–23. (The quote is on p. 19). Archived from the original on 9 August 2018; "access to Jahresbericht der Deutschen Mathematiker-Vereinigung". Archived from the original on 20 August 2017.

doi.org (Global: 2^nd place; English: 2^nd place)

Tao (2016), p. 68. Tao, Terence (2016). Analysis I. Texts and Readings in Mathematics. Vol. 37. Singapore: Springer. p. 68. doi:10.1007/978-981-10-1789-6. ISBN 978-981-10-1789-6.
Woodin, Greg; Winter, Bodo (2024). "Numbers in Context: Cardinals, Ordinals, and Nominals in American English". Cognitive Science. 48 (6) e13471. doi:10.1111/cogs.13471. PMC 11475258. PMID 38895756.
Tao (2016), p. 68 says, "... one of our main conceptualizations of natural numbers (is) that of cardinality, or measuring how many elements there are in a set". Tao, Terence (2016). Analysis I. Texts and Readings in Mathematics. Vol. 37. Singapore: Springer. p. 68. doi:10.1007/978-981-10-1789-6. ISBN 978-981-10-1789-6.
Tao (2016), p. 68: "Indeed, the Peano axiom approach treats natural numbers more like ordinals than cardinals." Tao, Terence (2016). Analysis I. Texts and Readings in Mathematics. Vol. 37. Singapore: Springer. p. 68. doi:10.1007/978-981-10-1789-6. ISBN 978-981-10-1789-6.
Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4 (1): 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856.
Baratella, Stefano; Ferro, Ruggero (1993). "A theory of sets with the negation of the axiom of infinity". Mathematical Logic Quarterly. 39 (3): 338–352. doi:10.1002/malq.19930390138. MR 1270381.
Kirby, Laurie; Paris, Jeff (1982). "Accessible Independence Results for Peano Arithmetic". Bulletin of the London Mathematical Society. 14 (4). Wiley: 285–293. doi:10.1112/blms/14.4.285. ISSN 0024-6093.

en.wikisource.org (Global: 82^nd place; English: 41^st place)

Poincaré, Henri (1905) [1902]. "On the nature of mathematical reasoning". La Science et l'hypothèse [Science and Hypothesis]. Translated by Greenstreet, William John. VI.

encyclopediaofmath.org (Global: 7,049^th place; English: 5,456^th place)

Mints, G. E. (ed.). "Peano axioms". Encyclopedia of Mathematics. Springer, in cooperation with the European Mathematical Society. Archived from the original on 13 October 2014. Retrieved 8 October 2014.

google.com.au (Global: 5,954^th place; English: 3,036^th place)

books.google.com.au

Ifrah, Georges (9 October 2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley. ISBN 978-0-471-39340-5.

hup.harvard.edu (Global: 6,606^th place; English: 4,067^th place)

von Neumann (1923). von Neumann, John (1923). "Zur Einführung der transfiniten Zahlen" [On the Introduction of the Transfinite Numbers]. Acta Litterarum AC Scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio Scientiarum Mathematicarum. 1: 199–208.
- von Neumann, John (January 2002) [1923]. "On the introduction of transfinite numbers". In van Heijenoort, Jean (ed.). From Frege to Gödel: A source book in mathematical logic, 1879–1931 (3rd ed.). Harvard University Press. pp. 346–354. ISBN 978-0-674-32449-7. – English translation of von Neumann (1923).

iso.org (Global: 1,190^th place; English: 1,097^th place)

"Standard number sets and intervals" (PDF). ISO 80000-2:2019 Quantities and units Part 2: Mathematics. International Organization for Standardization. 24 June 2025.

iteh.ai (Global: low place; English: low place)

cdn.standards.iteh.ai

"Standard number sets and intervals" (PDF). ISO 80000-2:2019 Quantities and units Part 2: Mathematics. International Organization for Standardization. 24 June 2025.

jstor.org (Global: 23^rd place; English: 15^th place)

Benacerraf (1965), p. 70: "To be the number 3 is no more and no less than to be preceded by 2, 1, and possibly 0, and to be followed by 4, 5, and so forth ... Any object can play the role of 3; that is, any object can be the third element in some progression. What is peculiar to 3 is that it defines that role – not by being a paradigm of any object which plays it, but by representing the relation that any third member of a progression bears to the rest of the progression." Benacerraf, Paul (January 1965). "What Numbers Could not Be". The Philosophical Review. 74: 47–73. JSTOR 2183530.
Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4 (1): 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856.

loc.gov (Global: 63^rd place; English: 47^th place)

lccn.loc.gov

Hamilton (1988), pp. 117 ff calls them "Peano's Postulates" and begins with "1. 0 is a natural number."
Halmos (1966), p. 46 uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I) $0 \in ω$ (where, of course, $0 = \emptyset$ " ( $ω$ is the set of all natural numbers).
Morash (1991), Section 10.1: An Axiomatization for the System of Positive Integers gives "a two-part axiom" in which the natural numbers begin with 1. Hamilton, A. G. (1988). Logic for Mathematicians (Revised ed.). Cambridge University Press. ISBN 978-0-521-36865-0. Halmos, Paul (1966). Naive Set Theory. Princeton: Van Nostrand. LCCN 60-11059. Morash, Ronald P. (1991). Bridge to Abstract Mathematics: Mathematical proof and structures (2nd ed.). Mcgraw-Hill. ISBN 978-0-07-043043-3.
Quine (1960), pp. 262–263, says the only thing required of an acceptable description of natural numbers is that they form a progression and so "any progression – i.e., any infinite series each of whose members has only finitely many precursors – will do nicely". Quine, Willard (1960). Word and Object. Technology Press (MIT). LCCN 60-9621.

msu.edu (Global: 1,979^th place; English: 1,265^th place)

archive.lib.msu.edu

"Natural Number". archive.lib.msu.edu.

nih.gov (Global: 5^th place; English: 5^th place)

ncbi.nlm.nih.gov

Woodin, Greg; Winter, Bodo (2024). "Numbers in Context: Cardinals, Ordinals, and Nominals in American English". Cognitive Science. 48 (6) e13471. doi:10.1111/cogs.13471. PMC 11475258. PMID 38895756.

pubmed.ncbi.nlm.nih.gov

Woodin, Greg; Winter, Bodo (2024). "Numbers in Context: Cardinals, Ordinals, and Nominals in American English". Cognitive Science. 48 (6) e13471. doi:10.1111/cogs.13471. PMC 11475258. PMID 38895756.

st-andrews.ac.uk (Global: 1,053^rd place; English: 1,274^th place)

mathshistory.st-andrews.ac.uk

"Earliest Known Uses of Some of the Words of Mathematics (N)". Maths History.

web.archive.org (Global: 1^st place; English: 1^st place)

Ganssle, Jack G. & Barr, Michael (2003). "integer". Embedded Systems Dictionary. Taylor & Francis. pp. 138 (integer), 247 (signed integer), & 276 (unsigned integer). ISBN 978-1-57820-120-4. Archived from the original on 29 March 2017. Retrieved 28 March 2017 – via Google Books.
Mints, G. E. (ed.). "Peano axioms". Encyclopedia of Mathematics. Springer, in cooperation with the European Mathematical Society. Archived from the original on 13 October 2014. Retrieved 8 October 2014.
Gray, Jeremy (2008). Plato's Ghost: The modernist transformation of mathematics. Princeton University Press. p. 153. ISBN 978-1-4008-2904-0. Archived from the original on 29 March 2017 – via Google Books.
Weber, Heinrich L. (1891–1892). "Kronecker". Jahresbericht der Deutschen Mathematiker-Vereinigung [Annual report of the German Mathematicians Association]. pp. 2:5–23. (The quote is on p. 19). Archived from the original on 9 August 2018; "access to Jahresbericht der Deutschen Mathematiker-Vereinigung". Archived from the original on 20 August 2017.

wolfram.com (Global: 1,091^st place; English: 1,169^th place)

mathworld.wolfram.com

Weisstein, Eric W. "Natural Number". MathWorld. Retrieved 11 August 2020.
Weisstein, Eric W. "Counting Number". MathWorld.

functions.wolfram.com

"Listing of the Mathematical Notations used in the Mathematical Functions Website: Numbers, variables, and functions". functions.wolfram.com. Retrieved 27 July 2020.

worldcat.org (Global: 4^th place; English: 4^th place)

search.worldcat.org

Kirby, Laurie; Paris, Jeff (1982). "Accessible Independence Results for Peano Arithmetic". Bulletin of the London Mathematical Society. 14 (4). Wiley: 285–293. doi:10.1112/blms/14.4.285. ISSN 0024-6093.

Natural number (English Wikipedia)

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hup.harvard.edu (Global: 6,606th place; English: 4,067th place)

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functions.wolfram.com

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ams.org (Global: 434^th place; English: 249^th place)

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hup.harvard.edu (Global: 6,606^th place; English: 4,067^th place)

iso.org (Global: 1,190^th place; English: 1,097^th place)

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