Natural number (English Wikipedia)

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

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19books.google.com

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8archive.org

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ams.org (Global: 451^st place; English: 277^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)

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 (in French). Paris, Gauthier-Villars. p. 39.
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? (in German). F. Vieweg. 1893. 71–73.
Arithmetices principia: nova methodo (in Latin). Fratres Bocca. 1889. p. 12.
Peano, Giuseppe (1901). Formulaire des mathematiques (in French). Paris, Gauthier-Villars. p. 39.

bnf.fr (Global: 124^th place; English: 544^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 (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 (1974, 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) gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1: An Axiomatization for the System of Positive Integers) Hamilton, A.G. (1988). Logic for Mathematicians (Revised ed.). Cambridge University Press. ISBN 978-0-521-36865-0 – via Google Books. Halmos, Paul (1974). Naive Set Theory. Springer Science & Business Media. ISBN 978-0-387-90092-6 – via Google Books. Morash, Ronald P. (1991). Bridge to Abstract Mathematics: Mathematical proof and structures (Second ed.). Mcgraw-Hill College. ISBN 978-0-07-043043-3 – via Google Books.
"Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." (Eves 1990, p. 606) Eves, Howard (1990). An Introduction to the History of Mathematics (6th ed.). Thomson. ISBN 978-0-03-029558-4 – via Google Books.
Cooke, Heather (26 October 2000). Primary Mathematics. SAGE. p. 14. ISBN 978-1-84787-949-3.
Zegarelli, Mark (28 January 2014). Basic Math and Pre-Algebra For Dummies. John Wiley & Sons. 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." Mendelson, Elliott (2008) [1973]. Number Systems and the Foundations of Analysis. Dover Publications. ISBN 978-0-486-45792-5 – via Google Books.
Bluman (2010, p. 1): "Numbers make up the foundation of mathematics." Bluman, Allan (2010). Pre-Algebra DeMYSTiFieD (Second ed.). McGraw-Hill Professional. ISBN 978-0-07-174251-1 – via Google Books.
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.
Eves 1990, Chapter 15 Eves, Howard (1990). An Introduction to the History of Mathematics (6th ed.). Thomson. ISBN 978-0-03-029558-4 – 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: 5,626^th place; English: 8,880^th 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)

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, Terence (2016). Analysis I. Texts and Readings in Mathematics. Vol. 37. Singapore: Springer Singapore. 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.

encyclopediaofmath.org (Global: 3,863^rd place; English: 2,637^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: 2,205^th place; English: 4,412^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.

gutenberg.org (Global: 489^th place; English: 377^th place)

Russell, Bertrand (18 December 2012). Introduction to Mathematical Philosophy.

iso.org (Global: 629^th place; English: 610^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: 26^th place; English: 20^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 – via JSTOR.
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.

msu.edu (Global: 1,844^th place; English: 1,231^st place)

archive.lib.msu.edu

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

nih.gov (Global: 4^th place; English: 4^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.

springer.com (Global: 274^th place; English: 309^th place)

link.springer.com

Tao, Terence (2016). Analysis I. Texts and Readings in Mathematics. Vol. 37. Singapore: Springer Singapore. doi:10.1007/978-981-10-1789-6. ISBN 978-981-10-1789-6.

st-andrews.ac.uk (Global: 1,547^th place; English: 1,410^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.

wikisource.org (Global: 27^th place; English: 51^st place)

en.wikisource.org

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

wolfram.com (Global: 513^th place; English: 537^th place)

mathworld.wolfram.com

Weisstein, Eric W. "Natural Number". mathworld.wolfram.com. 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: 5^th place; English: 5^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)

ams.org (Global: 451st place; English: 277th place)

mathscinet.ams.org

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

bnf.fr (Global: 124th place; English: 544th place)

gallica.bnf.fr

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

digizeitschriften.de (Global: 5,626th place; English: 8,880th place)

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

encyclopediaofmath.org (Global: 3,863rd place; English: 2,637th place)

google.com.au (Global: 2,205th place; English: 4,412th place)