Editing Natural number (section)

===Formal construction===

In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. [[Henri Poincaré]] stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act.<ref>{{cite book |last1=Poincaré |first1=Henri|translator1-first=William John|translator1-last= Greenstreet |title=La Science et l'hypothèse|trans-title=Science and Hypothesis|orig-date=1902|date=1905|chapter=On the nature of mathematical reasoning|chapter-url=https://en.wikisource.org/wiki/Science_and_Hypothesis/Chapter_1|at=VI}}</ref> [[Leopold Kronecker]] summarized his belief as "God made the integers, all else is the work of man".{{efn|The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891–1892, 19, quoting from a lecture of Kronecker's of 1886."<ref>{{cite book
 |last=Gray |first=Jeremy |author-link=Jeremy Gray
 |year=2008
 |title=Plato's Ghost: The modernist transformation of mathematics
 |page=153
 |publisher=Princeton University Press
 |isbn=978-1-4008-2904-0
 |via=Google Books
 |url=https://books.google.com/books?id=ldzseiuZbsIC&q=%22God+made+the+integers%2C+all+else+is+the+work+of+man.%22
 |url-status=live
 |archive-url=https://web.archive.org/web/20170329150904/https://books.google.com/books?id=ldzseiuZbsIC&q=%22God+made+the+integers%2C+all+else+is+the+work+of+man.%22#v=snippet&q=%22God%20made%20the%20integers%2C%20all%20else%20is%20the%20work%20of%20man.%22&f=false
 |archive-date=29 March 2017
}}</ref><ref>{{cite book
 |last=Weber |first=Heinrich L.
 |year=1891–1892
 |chapter=Kronecker
 |chapter-url=http://www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6 
 |archive-url=https://web.archive.org/web/20180809110042/http://www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6
 |archive-date=9 August 2018
 |title=''Jahresbericht der Deutschen Mathematiker-Vereinigung''
 |trans-title=Annual report of the German Mathematicians Association
 |pages=2:5–23. (The quote is on p.&nbsp;19)
 |postscript=;
}} {{cite web
 |title=access to ''Jahresbericht der Deutschen Mathematiker-Vereinigung''
 |url=http://www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002
 |archive-url=https://web.archive.org/web/20170820201100/http://www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002
 |archive-date=20 August 2017 
}}</ref>}}

The [[Constructivism (mathematics)|constructivists]] saw a need to improve upon the logical rigor in the [[foundations of mathematics]].{{efn|"Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." {{harv|Eves|1990|p=606}} }} In the 1860s, [[Hermann Grassmann]] suggested a [[recursive definition]] for natural numbers, thus stating they were not really natural—but a consequence of definitions. Later, two classes of such formal definitions emerged, using set theory and Peano's axioms respectively. Later still, they were shown to be equivalent in most practical applications.

[[Set-theoretical definitions of natural numbers]] were initiated by [[Gottlob Frege|Frege]]. He initially defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set. However, this definition turned out to lead to paradoxes, including [[Russell's paradox]]. To avoid such paradoxes, the formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements.<ref>{{harvnb|Eves|1990|loc=Chapter 15}}</ref>

In 1881, [[Charles Sanders Peirce]] provided the first [[Axiomatic system#Axiomatization|axiomatization]] of natural-number arithmetic.<ref>{{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}}</ref><ref>{{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}}</ref> In 1888, [[Richard Dedekind]] proposed another axiomatization of natural-number arithmetic,<ref>{{cite book |title=Was sind und was sollen die Zahlen? |date=1893 |publisher=F. Vieweg |url=https://archive.org/details/wassindundwasso00dedegoog/page/n42/mode/2up |language=German|at=71–73}}</ref> and in&nbsp;1889, Peano published a simplified version of Dedekind's axioms in his book ''The principles of arithmetic presented by a new method'' ({{langx|la|[[Arithmetices principia, nova methodo exposita]]}}). This approach is now called [[Peano arithmetic]]. It is based on an [[axiomatization]] of the properties of [[ordinal number]]s: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is [[equiconsistent]] with several weak systems of [[set theory]]. One such system is [[ZFC]] with the [[axiom of infinity]] replaced by its negation.<ref>{{cite journal
 | last1 = Baratella | first1 = Stefano
 | last2 = Ferro | first2 = Ruggero
 | doi = 10.1002/malq.19930390138
 | issue = 3
 | journal = Mathematical Logic Quarterly
 | mr = 1270381
 | pages = 338–352
 | title = A theory of sets with the negation of the axiom of infinity
 | volume = 39
 | year = 1993}}</ref> Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include [[Goodstein's theorem]].<ref>{{cite journal | last1=Kirby | first1=Laurie | last2=Paris | first2=Jeff | title=Accessible Independence Results for Peano Arithmetic | journal=Bulletin of the London Mathematical Society | publisher=Wiley | volume=14 | issue=4 | year=1982 | issn=0024-6093 | doi=10.1112/blms/14.4.285 | pages=285–293}}</ref>