Natural numbers: Difference between revisions
added algebraic properties
(Created page with "The '''natural numbers''', or '''counting numbers''', are a system of numbers which includes the positive integers \( 1, 2, 3, \dots \), and under some definitions also includes zero. If zero is to be considered a natural number, which is usually the case in set theory, the natural numbers are precisely the finite ordinals. ==Encodings== The natural numbers are fundamental objects in mathematics, and thus different areas of math have different conventions of e...") |
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==Encodings==
The natural numbers are fundamental objects in mathematics, and thus different areas of math have different conventions of encoding or defining them. In many areas such as geometry or number theory, they are taken as primitives and not formally treated
===Von Neumann ordinals===
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===Zermelo ordinals===
[[:wikipedia:Ernst Zermelo|Ernst Zermelo]] provided an alternative construction of the natural numbers, encoding \( 0 = \varnothing \) and \( n + 1 = \{ n \} \) for \( n \ge 0 \). Unlike the Von Neumann ordinals, Zermelo's encoding can only be used to represent finite ordinals.
===Frege and Russell===
During the early development of foundational philosophy and logicism, [[:wikipedia:Gottlob Frege|Gottlob Frege]] and [[:wikipedia:Bertrand Russell|Bertrand Russell]] proposed defining a natural number \( n \) as the equivalence [[class]] of all sets with [[cardinality]] \( n \). This definition cannot be realized in ZFC, because the classes involved are [[proper class]]es, except for \( n = 0 \).
===Church numerals===
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Axiomatic systems that describe properties of the naturals are called arithmetics. Two of the most popular are [[Peano arithmetic]] and [[second-order arithmetic]].
==Algebraic properties of the natural numbers==
The natural numbers (including zero) are closed under addition and multiplication. They satisfy commutativity and associativity of both operations, and distributivity of multiplication over addition. They form a monoid under addition, which is the free monoid with one generator. In addition, positive naturals form a monoid under multiplication -- the free monoid with countably infinite generators, which are the prime numbers.
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