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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:
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.