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# Natural number

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A natural number is any of the numbers 0, 1, 2, 3... that can be used to measure the size of finite sets.

While many authors exclude zero, which is the historical standard, most axiomatizations include it now. In this encyclopedia, zero is considered to be a natural number.

Though even a small child will understand what we mean by natural numbers, their definition has not been easy. The [Peano postulates]? essentially uniquely describe the set N of natural numbers:

- There is a natural number 0.
- Every natural number a has a successor, denoted by a + 1.
- There is no natural number whose successor is 0.
- Distinct natural numbers have distinct successors: if a <> b, then a + 1 <> b + 1.
- If a property is possessed by 0 and and also by the successor of every natural number it is possessed by, then it is possessed by all natural numbers.

The last postulate ensures that the proof technique of mathematical induction is valid.

A standard construction in set theory is to define each natural number as the set of natural numbers less than it, so that 0 = {}, 1 = {0}, 2 = {0,1}, 3 = {0,1,2}... when you see a natural number used as a set this is typically what is meant.

Natural numbers can be used for two purposes: describing the position of an element in an ordered sequence, which is generalized by the concept of ordinal number, and counting the elements of a set, which is generalized by the concept of [cardinal number]?. In the finite world, these two concepts coincide: the finite ordinals are equal to N as are the finite cardinals. When moving beyond the finite however, the two concepts diverge.

One can inductively define an addition on the natural numbers by requiring a + (b + 1) = (a + b) + 1. This turns the natural numbers (N,+) into a commutative monoid with neutral element 0, the so-called free monoid with one generator. This monoid satisfies the cancellation property and can therefore be embedded in a group. The smallest group containing the natural numbers is the integers.

Furthermore, one can define a total order on the natural numbers by writing a <= b if and only if there exists another natural number c with a + c = b.

The properties of the natural numbers are studied in number theory.

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Last edited August 12, 2001 7:20 am (diff)
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