Natural number (English Wikipedia)

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

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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

Emerson, William (1763). The method of increments. p. 113.
Arithmetices principia: nova methodo (in Latin). Fratres Bocca. 1889. p. 12.
Peano, Giuseppe (1901). Formulaire des mathematiques (in French). Paris, Gauthier-Villars. p. 39.
Goldrei, Derek (1998). "3". Classic Set Theory: A guided independent study (1. ed., 1. print ed.). Boca Raton, Fla. [u.a.]: Chapman & Hall/CRC. p. 33. ISBN 978-0-412-60610-6.
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.
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.
Rudin, W. (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. p. 25. ISBN 978-0-07-054235-8.

arxiv.org

Fletcher, Peter; Hrbacek, Karel; Kanovei, Vladimir; Katz, Mikhail G.; Lobry, Claude; Sanders, Sam (2017). "Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others". Real Analysis Exchange. 42 (2): 193–253. arXiv:1703.00425. doi:10.14321/realanalexch.42.2.0193.

astronomicalheritage.net

www2.astronomicalheritage.net

"The Ishango Bone, Democratic Republic of the Congo". UNESCO's Portal to the Heritage of Astronomy. Archived from the original on 10 November 2014., on permanent display at the Royal Belgian Institute of Natural Sciences, Brussels, Belgium.

bnf.fr

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

Mac Lane & Birkhoff (1999, p. 15) include zero in the natural numbers: 'Intuitively, the set $\mathbb {N} =\{0,1,2,\ldots \}$ of all natural numbers may be described as follows: $\mathbb {N}$ contains an "initial" number $0$ ; ...'. They follow that with their version of the Peano's axioms. Mac Lane, Saunders; Birkhoff, Garrett (1999). Algebra (3rd ed.). American Mathematical Society. ISBN 978-0-8218-1646-2 – 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.
Hamilton (1988, pp. 117 ff) calls them "Peano's Postulates" and begins with "1. 0 is a natural number."
Halmos (1960, 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 (1960). Naive Set Theory. Springer Science & Business Media. ISBN 978-0-387-90092-6 – via Google Books. {{cite book}}: ISBN / Date incompatibility (help) 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.
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.
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.
Mann, Charles C. (2005). 1491: New Revelations of the Americas before Columbus. Knopf. p. 19. ISBN 978-1-4000-4006-3. Archived from the original on 14 May 2015. Retrieved 3 February 2015 – via Google Books.
Evans, Brian (2014). "Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations". The Development of Mathematics Throughout the Centuries: A brief history in a cultural context. John Wiley & Sons. ISBN 978-1-118-85397-9 – via Google Books.
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.
Křížek, Michal; Somer, Lawrence; Šolcová, Alena (21 September 2021). From Great Discoveries in Number Theory to Applications. Springer Nature. p. 6. ISBN 978-3-030-83899-7.
See, for example, Carothers (2000, p. 3) or Thomson, Bruckner & Bruckner (2008, p. 2) Carothers, N.L. (2000). Real Analysis. Cambridge University Press. ISBN 978-0-521-49756-5 – via Google Books. Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008). Elementary Real Analysis (Second ed.). ClassicalRealAnalysis.com. ISBN 978-1-4348-4367-8 – via Google Books.
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.
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.

clarku.edu

aleph0.clarku.edu

Euclid. "Book VII, definitions 1 and 2". In Joyce, D. (ed.). Elements. Clark University.
Euclid. "Book VII, definition 22". In Joyce, D. (ed.). Elements. Clark University. A perfect number is that which is equal to the sum of its own parts. In definition VII.3 a "part" was defined as a number, but here 1 is considered to be a part, so that for example $6 = 1 + 2 + 3$ is a perfect number.

digizeitschriften.de

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

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.
Brown, Jim (1978). "In defense of index origin 0". ACM SIGAPL APL Quote Quad. 9 (2): 7. doi:10.1145/586050.586053. S2CID 40187000.
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.
Fletcher, Peter; Hrbacek, Karel; Kanovei, Vladimir; Katz, Mikhail G.; Lobry, Claude; Sanders, Sam (2017). "Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others". Real Analysis Exchange. 42 (2): 193–253. arXiv:1703.00425. doi:10.14321/realanalexch.42.2.0193.

encyclopediaofmath.org

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.

fyx.hu

acta.fyx.hu

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. Archived from the original on 18 December 2014. Retrieved 15 September 2013.

iso.org

ISO 80000-2:2019 Quantities and units Part 2: Mathematics. International Organization for Standardization. 24 June 2025.

jsoftware.com

Hui, Roger. "Is index origin 0 a hindrance?". jsoftware.com. Archived from the original on 20 October 2015. Retrieved 19 January 2015.

jstor.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.

merriam-webster.com

"natural number". Merriam-Webster.com. Merriam-Webster. Archived from the original on 13 December 2019. Retrieved 4 October 2014.

msu.edu

archive.lib.msu.edu

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

msu.ru

hbar.phys.msu.ru

Deckers, Michael (25 August 2003). "Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius". Hbar.phys.msu.ru. Archived from the original on 15 January 2019. Retrieved 13 February 2012.

naturalsciences.be

"Introduction". Ishango bone. Brussels, Belgium: Royal Belgian Institute of Natural Sciences. Archived from the original on 4 March 2016.

ishango.naturalsciences.be

"Flash presentation". Ishango bone. Brussels, Belgium: Royal Belgian Institute of Natural Sciences. Archived from the original on 27 May 2016.

nih.gov

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.

semanticscholar.org

api.semanticscholar.org

Brown, Jim (1978). "In defense of index origin 0". ACM SIGAPL APL Quote Quad. 9 (2): 7. doi:10.1145/586050.586053. S2CID 40187000.

st-and.ac.uk

www-history.mcs.st-and.ac.uk

"A history of Zero". MacTutor History of Mathematics. Archived from the original on 19 January 2013. Retrieved 23 January 2013.

st-andrews.ac.uk

mathshistory.st-andrews.ac.uk

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

stanford.edu

plato.stanford.edu

Bagaria, Joan (2017). Set Theory (Winter 2014 ed.). The Stanford Encyclopedia of Philosophy. Archived from the original on 14 March 2015. Retrieved 13 February 2015.

web.archive.org

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.
"Introduction". Ishango bone. Brussels, Belgium: Royal Belgian Institute of Natural Sciences. Archived from the original on 4 March 2016.
"Flash presentation". Ishango bone. Brussels, Belgium: Royal Belgian Institute of Natural Sciences. Archived from the original on 27 May 2016.
"The Ishango Bone, Democratic Republic of the Congo". UNESCO's Portal to the Heritage of Astronomy. Archived from the original on 10 November 2014., on permanent display at the Royal Belgian Institute of Natural Sciences, Brussels, Belgium.
"A history of Zero". MacTutor History of Mathematics. Archived from the original on 19 January 2013. Retrieved 23 January 2013.
Mann, Charles C. (2005). 1491: New Revelations of the Americas before Columbus. Knopf. p. 19. ISBN 978-1-4000-4006-3. Archived from the original on 14 May 2015. Retrieved 3 February 2015 – via Google Books.
Deckers, Michael (25 August 2003). "Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius". Hbar.phys.msu.ru. Archived from the original on 15 January 2019. Retrieved 13 February 2012.
Bagaria, Joan (2017). Set Theory (Winter 2014 ed.). The Stanford Encyclopedia of Philosophy. Archived from the original on 14 March 2015. Retrieved 13 February 2015.
"natural number". Merriam-Webster.com. Merriam-Webster. Archived from the original on 13 December 2019. Retrieved 4 October 2014.
Hui, Roger. "Is index origin 0 a hindrance?". jsoftware.com. Archived from the original on 20 October 2015. Retrieved 19 January 2015.
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.
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.
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. Archived from the original on 18 December 2014. Retrieved 15 September 2013.

wikisource.org

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

mathworld.wolfram.com

Weisstein, Eric W. "Natural Number". mathworld.wolfram.com. Retrieved 11 August 2020.
Weisstein, Eric W. "Counting Number". MathWorld.
Weisstein, Eric W. "Multiplication". mathworld.wolfram.com. Retrieved 27 July 2020.

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

search.worldcat.org

Mueller, Ian (2006). Philosophy of mathematics and deductive structure in Euclid's Elements. Mineola, New York: Dover Publications. p. 58. ISBN 978-0-486-45300-2. OCLC 69792712.
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.