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0 = { }

1 = {0} = {{ }}

2 = {1} = {{{ }}}, etc.

Or we could even define 0 = {{ }}

and S(a) = a U {a}

producing

0 = {{ }}

1 = {{ }, 0} = {{ }, {{ }}}

2 = {{ }, 0, 1}, etc.

Arguably the oldest set-theoretic definition of the natural numbers is the definition commonly ascribed to Frege and Russell under which each concrete natural number n is defined as the set of all sets with n elements. This may appear circular, but can be made rigorous with care. Define 0 as ${\displaystyle \{\{\}\}}$ (clearly the set of all sets with 0 elements) and define ${\displaystyle \sigma (A)}$ (for any set A) as ${\displaystyle \{x\cup \{y\}\mid x\in A\wedge y\not \in x\}}$. Then 0 will be the set of all sets with 0 elements, ${\displaystyle 1=\sigma (0)}$ will be the set of all sets with 1 element, ${\displaystyle 2=\sigma (1)}$ will be the set of all sets with 2 elements, and so forth. The set of all natural numbers can be defined as the intersection of all sets containing 0 as an element and closed under ${\displaystyle \sigma }$ (that is, if the set contains an element n, it also contains ${\displaystyle \sigma (n)}$). This definition does not work in the usual systems of axiomatic set theory because the collections involved are too large (it will not work in any set theory with the axiom of separation); but it does work in New Foundations (and in related systems known to be consistent) and in some systems of type theory.

For the rest of this article, we follow the standard construction described above.

Properties

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

If we define 1 := S(0), then b + 1 = b + S(0) = S(b + 0) = S(b). That is, b + 1 is simply the successor of b.

Analogously, given that addition has been defined, a multiplication × can be defined via a × 0 = 0 and a × S(b) = (a × b) + a. This turns (N, ×) into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers. Addition and multiplication are compatible, which is expressed in the distribution law: a × (b + c) = (a × b) + (a × c). These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative.

If we interpret the natural numbers as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that we start with a + 1 = S(a) and a × 1 = a.

For the remainder of the article, we write ab to indicate the product a × b, and we also assume the standard order of operations.

Furthermore, one defines 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. This order is compatible with the arithmetical operations in the following sense: if a, b and c are natural numbers and a ≤ b, then a + c ≤ b + c and ac ≤ bc. An important property of the natural numbers is that they are well-ordered: every non-empty set of natural numbers has a least element.

While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder is available as a substitute: for any two natural numbers a and b with b ≠ 0 we can find natural numbers q and r such that

a = bq + r and r < b

The number q is called the quotient and r is called the remainder of division of a by b. The numbers q and r are uniquely determined by a and b. This, the Division algorithm, is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.

The natural numbers including zero form a commutative monoid under addition (with identity element zero), and under multiplication (with identity element one).

Generalizations

Two generalizations of natural numbers arise from the two uses: ordinal numbers are used to describe the position of an element in an ordered sequence and cardinal numbers are used to specify the size of a given set.

For finite sequences or finite sets, both of these properties are embodied in the natural numbers.

Other generalizations are discussed in the article on numbers.

References

• Edmund Landau, Foundations of Analysis, Chelsea Pub Co. ISBN 0-8218-2693-X.

External links

• Axioms and Construction of Natural Numbers

• Cardinal numbers
• Elementary mathematics
• Integers
• Number theory
• Numbers