Editing Natural number (section)

===Peano axioms===
{{Main|Peano axioms}}

The five Peano axioms are the following:<ref>{{cite encyclopedia
 |editor-first=G.E. |editor-last=Mints
 |title=Peano axioms
 |encyclopedia=Encyclopedia of Mathematics
 |publisher=[[Springer Science+Business Media|Springer]], in cooperation with the [[European Mathematical Society]]
 |url=http://www.encyclopediaofmath.org/index.php/Peano_axioms
 |url-status=live |access-date=8 October 2014
|archive-url=https://web.archive.org/web/20141013163028/http://www.encyclopediaofmath.org/index.php/Peano_axioms
 |archive-date=13 October 2014
}}</ref>{{efn|{{harvtxt|Hamilton|1988|pages=117&nbsp;ff}} calls them "Peano's Postulates" and begins with "1.{{spaces|2}}0 is a natural number."<br/>
{{harvtxt|Halmos|1960|page=46}} uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I){{spaces|2}}{{math|0 ∈ ω}} (where, of course, {{math|0 {{=}} ∅}}" ({{math|ω}} is the set of all natural numbers).<br/>
{{harvtxt|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'')
}}

# 0 is a natural number.
# Every natural number has a successor which is also a natural number.
# 0 is not the successor of any natural number.
# If the successor of <math> x </math> equals the successor of <math> y </math>, then <math> x</math> equals <math> y</math>.
# The [[axiom of induction]]: If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.

These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic, the successor of <math> x</math> is <math> x + 1</math>.