In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a mathematical structure that satisfies the definition.
The article presents several such constructions. They are equivalent in the sense that, given the result of any two such constructions, there is a unique isomorphism of ordered field between them. This results from the above definition and is independent of particular constructions. These isomorphisms allow identifying the results of the constructions, and, in practice, to forget which construction has been chosen.
Axiomatic definitions
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. This means the following. The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real numbers and denoted respectively with and ; the binary relation is inequality, denoted Moreover, the following properties called axioms must be satisfied.
The existence of such a structure is a theorem, which is proved by constructing such a structure. A consequence of the axioms is that this structure is unique up to an isomorphism, and thus, the real numbers can be used and manipulated, without referring to the method of construction.
Axioms
is a field under addition and multiplication. In other words, For all x, y, and z in , x + (y + z) = (x + y) + z and x × (y × z) = (x × y) × z. (associativity of addition and multiplication) For all x and y in , x + y = y + x and x × y = y × x. (commutativity of addition and multiplication) For all x, y, and z in , x × (y + z) = (x × y) + (x × z). (distributivity of multiplication over addition) For all x in , x + 0 = x. (existence of additive identity) 0 is not equal to 1, and for all x in , x × 1 = x. (existence of multiplicative identity) For every x in , there exists an element −x in , such that x + (−x) = 0. (existence of additive inverses) For every x ≠ 0 in , there exists an element x−1 in , such that x × x−1 = 1. (existence of multiplicative inverses) is totally ordered for . In other words, For all x in , x ≤ x. (reflexivity) For all x and y in , if x ≤ y and y ≤ x, then x = y. (antisymmetry) For all x, y, and z in , if x ≤ y and y ≤ z, then x ≤ z. (transitivity) For all x and y in , x ≤ y or y ≤ x. (totality) Addition and multiplication are compatible with the order. In other words, For all x, y and z in , if x ≤ y, then x + z ≤ y + z. (preservation of order under addition) For all x and y in , if 0 ≤ x and 0 ≤ y, then 0 ≤ x × y (preservation of order under multiplication) The order ≤ is complete in the following sense: every non-empty subset of that is bounded above has a least upper bound. In other words, If A is a non-empty subset of , and if A has an upper bound in then A has a least upper bound u, such that for every upper bound v of A, u ≤ v.
On the least upper bound property
Axiom 4, which requires the order to be Dedekind-complete, implies the Archimedean property.
The axiom is crucial in the characterization of the reals. For example, the totally ordered field of the rational numbers Q satisfy the first three axioms, but not the fourth. In other words, models of the rational numbers are also models of the first three axioms.
Note that the axiom is not firstorderizable, as it expresses a statement about collections of reals and not just individual such numbers. As such, the reals are not given by a first-order logic theory.
On models
A model of real numbers is a mathematical structure that satisfies the above axioms. Several models are given below. Any two models are isomorphic; so, the real numbers are unique up to isomorphisms.
Saying that any two models are isomorphic means that for any two models and there is a bijection that preserves both the field operations and the order. Explicitly, is both injective and surjective. and . and , for all and in if and only if , for all and in