(mathbb{C}) denotes the set of complex numbers. (This sentence will be introduced more formally later.) For example, (mathbb{Z}^3) is the set of all tuples (3) whose entries are integers. In other words, (mathbb{Z}^3 = left { begin{bmatrix} a b cend{bmatrix} : a, b, c in mathbb{Z}right}). The following definition was first used by John von Neumann,[36] although Levy traces the idea back to unpublished works by Zermelo from 1916. [37] Since this definition extends to infinite sets as the definition of the ordinal, the sets considered below are sometimes called von Neumann ordinals. A set is a well-defined collection of disparate mathematical objects. Objects are called members or members of the set. If the natural numbers are considered to be “without 0” and “starting at 1”, the definitions of + and × are as above, except that they start with a + 1 = S (a) and a × 1 = a. Also, ( N ∗ , + ) {displaystyle (mathbb {N^{*}} ,+)} has no identity element. With this definition, for a natural number n, the phrase “a set S has n elements” can be formally defined as “there exists a bijection from n to S. This formalizes the operation of counting the elements of S. Also, ≤ m if and only if n is a subset of m. In other words, the inclusion of sets defines the usual global order on the natural numbers.

This order is a good order. Since natural numbers naturally form a subset of integers (often called Z {displaystyle mathbb {Z} }), they can also be called positive or non-negative integers. [30] To be clear whether 0 is included or not, an index (or exponent) “0” is sometimes added in the first case and an exponent “*” in the second:[3] The second definition is based on set theory. It defines natural numbers as specific sets. More precisely, any natural number n is defined as an explicitly defined set whose elements allow to count the elements of other sets, in the sense that the phrase “a set S has n elements” means that there is a univocal correspondence between the two sets n and S. (a in A) means that (a) is a member of (A). Example: (5 in mathbb{Q}) For the set N {displaystyle mathbb {N} } the natural numbers and the successor function S : N → N {displaystyle Scolon mathbb {N} to mathbb {N} }, which sends each natural number to the next, one can define the addition of the natural numbers recursively by defining a + 0 = a and a + S(b) = S(a + b) for all a, b. Then ( N , + ) {displaystyle (mathbb {N} ,+)} is a commutative monoid with the identity element 0.

It is a free monoid on a generator. This commutative monoid satisfies the undo property so that it can be incorporated into a group. The smallest group containing the natural numbers is the integers. The theoretical definitions of natural numbers were initiated by Frege. He first defined a natural number as the class of all sets that are uniquely matched to a given set. However, this definition has led to paradoxes, including Russell`s paradox. To avoid such paradoxes, the formalism has been modified so that a natural number is defined as a certain set, and any set that can be brought in unambiguous correspondence with this set is said to have this number of elements. [20] The set of all natural numbers is denoted N or N by default. {displaystyle mathbb {N} .} [1] [27] Older texts sometimes used J as a symbol for this set.

[28] The sets used to define the natural numbers satisfy Peano`s axioms. It follows that any theorem that can be established and proved in Peano arithmetic can also be proved in set theory. However, the two definitions are not equivalent because there are theorems that can be expressed with respect to Peano arithmetic and proved in set theory that are not provable in Peano arithmetic. A likely example is Fermat`s last sentence. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind and studied by Giuseppe Peano; This approach is now called Peano arithmetic. It is based on an axiomatization of the properties of ordinal numbers: every natural number has a successor and every nonzero natural number has a unique predecessor. Peano arithmetic is synonymous with several weak systems of set theory.