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.^{ [1]} 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.
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field.^{ [2]}^{ [3]}^{ [4]} 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.
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.
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,
An alternative synthetic axiomatization of the real numbers and their arithmetic was given by Alfred Tarski, consisting of only the 8 axioms shown below and a mere four primitive notions: a set called the real numbers, denoted , a binary relation over called order, denoted by infix <, a binary operation over called addition, denoted by infix +, and the constant 1.
Axioms of order (primitives: , <):
Axiom 1. If x < y, then not y < x. That is, "<" is an asymmetric relation.
Axiom 2. If x < z, there exists a y such that x < y and y < z. In other words, "<" is dense in .
Axiom 3. "<" is Dedekind-complete. More formally, for all X, Y ⊆ , if for all x ∈ X and y ∈ Y, x < y, then there exists a z such that for all x ∈ X and y ∈ Y, if z ≠ x and z ≠ y, then x < z and z < y.
To clarify the above statement somewhat, let X ⊆ and Y ⊆ . We now define two common English verbs in a particular way that suits our purpose:
Axiom 3 can then be stated as:
Axioms of addition (primitives: , <, +):
Axiom 4. x + (y + z) = (x + z) + y.
Axiom 5. For all x, y, there exists a z such that x + z = y.
Axiom 6. If x + y < z + w, then x < z or y < w.
Axioms for one (primitives: , <, +, 1):
Axiom 7. 1 ∈ .
Axiom 8. 1 < 1 + 1.
These axioms imply that is a linearly ordered abelian group under addition with distinguished element 1. is also Dedekind-complete and divisible.
We shall not prove that any models of the axioms are isomorphic. Such a proof can be found in any number of modern analysis or set theory textbooks. We will sketch the basic definitions and properties of a number of constructions, however, because each of these is important for both mathematical and historical reasons. The first three, due to Georg Cantor/ Charles Méray, Richard Dedekind/ Joseph Bertrand and Karl Weierstrass all occurred within a few years of each other. Each has advantages and disadvantages. A major motivation in all three cases was the instruction of mathematics students.
A standard procedure to force all Cauchy sequences in a metric space to converge is adding new points to the metric space in a process called completion.
is defined as the completion of Q with respect to the metric |x-y|, as will be detailed below (for completions of Q with respect to other metrics, see p-adic numbers.)
Let R be the set of Cauchy sequences of rational numbers. That is, sequences
of rational numbers such that for every rational ε > 0, there exists an integer N such that for all natural numbers m,n > N, |x_{m} − x_{n}| < ε. Here the vertical bars denote the absolute value.
Cauchy sequences (x_{n}) and (y_{n}) can be added and multiplied as follows:
Two Cauchy sequences are called equivalent if and only if the difference between them tends to zero. This defines an equivalence relation that is compatible with the operations defined above, and the set R of all equivalence classes can be shown to satisfy all axioms of the real numbers. We can embed Q into R by identifying the rational number r with the equivalence class of the sequence (r,r,r, …).
Comparison between real numbers is obtained by defining the following comparison between Cauchy sequences: (x_{n}) ≥ (y_{n}) if and only if x is equivalent to y or there exists an integer N such that x_{n} ≥ y_{n} for all n > N.
By construction, every real number x is represented by a Cauchy sequence of rational numbers. This representation is far from unique; every rational sequence that converges to x is a representation of x. This reflects the observation that one can often use different sequences to approximate the same real number.^{ [5]}
The only real number axiom that does not follow easily from the definitions is the completeness of ≤, i.e. the least upper bound property. It can be proved as follows: Let S be a non-empty subset of R and U be an upper bound for S. Substituting a larger value if necessary, we may assume U is rational. Since S is non-empty, we can choose a rational number L such that L < s for some s in S. Now define sequences of rationals (u_{n}) and (l_{n}) as follows:
For each n consider the number:
If m_{n} is an upper bound for S set:
Otherwise set:
This defines two Cauchy sequences of rationals, and so we have real numbers l = (l_{n}) and u = (u_{n}). It is easy to prove, by induction on n that:
and:
Thus u is an upper bound for S. To see that it is a least upper bound, notice that the limit of (u_{n} − l_{n}) is 0, and so l = u. Now suppose b < u = l is a smaller upper bound for S. Since (l_{n}) is monotonic increasing it is easy to see that b < l_{n} for some n. But l_{n} is not an upper bound for S and so neither is b. Hence u is a least upper bound for S and ≤ is complete.
The usual decimal notation can be translated to Cauchy sequences in a natural way. For example, the notation π = 3.1415... means that π is the equivalence class of the Cauchy sequence (3, 3.1, 3.14, 3.141, 3.1415, ...). The equation 0.999... = 1 states that the sequences (0, 0.9, 0.99, 0.999,...) and (1, 1, 1, 1,...) are equivalent, i.e., their difference converges to 0.
An advantage of constructing R as the completion of Q is that this construction is not specific to one example; it is used for other metric spaces as well.
A Dedekind cut in an ordered field is a partition of it, (A, B), such that A is nonempty and closed downwards, B is nonempty and closed upwards, and A contains no greatest element. Real numbers can be constructed as Dedekind cuts of rational numbers.^{ [6]}^{ [7]}
For convenience we may take the lower set as the representative of any given Dedekind cut , since completely determines . By doing this we may think intuitively of a real number as being represented by the set of all smaller rational numbers. In more detail, a real number is any subset of the set of rational numbers that fulfills the following conditions:^{ [8]}
As an example of a Dedekind cut representing an irrational number, we may take the positive square root of 2. This can be defined by the set .^{ [9]} It can be seen from the definitions above that is a real number, and that . However, neither claim is immediate. Showing that is real requires showing that has no greatest element, i.e. that for any positive rational with , there is a rational with and The choice works. Then but to show equality requires showing that if is any rational number with , then there is positive in with .
An advantage of this construction is that each real number corresponds to a unique cut. Furthermore, by relaxing the first two requirements of the definition of a cut, the extended real number system may be obtained by associating with the empty set and with all of .
As in the hyperreal numbers, one constructs the hyperrationals ^{*}Q from the rational numbers by means of an ultrafilter.^{ [10]}^{ [11]} Here a hyperrational is by definition a ratio of two hyperintegers. Consider the ring B of all limited (i.e. finite) elements in ^{*}Q. Then B has a unique maximal ideal I, the infinitesimal numbers. The quotient ring B/I gives the field R of real numbers^{[ citation needed]}. Note that B is not an internal set in ^{*}Q. Note that this construction uses a non-principal ultrafilter over the set of natural numbers, the existence of which is guaranteed by the axiom of choice.
It turns out that the maximal ideal respects the order on ^{*}Q. Hence the resulting field is an ordered field. Completeness can be proved in a similar way to the construction from the Cauchy sequences.
Every ordered field can be embedded in the surreal numbers. The real numbers form a maximal subfield that is Archimedean (meaning that no real number is infinitely large or infinitely small). This embedding is not unique, though it can be chosen in a canonical way.
A relatively less known construction allows to define real numbers using only the additive group of integers with different versions.^{ [12]}^{ [13]}^{ [14]} The construction has been formally verified by the IsarMathLib project.^{ [15]} Shenitzer (1987) and Arthan (2004) refer to this construction as the Eudoxus reals, named after an ancient Greek astronomer and mathematician Eudoxus of Cnidus.
Let an almost homomorphism be a map such that the set is finite. (Note that is an almost homomorphism for every .) Almost homomorphisms form an abelian group under pointwise addition. We say that two almost homomorphisms are almost equal if the set is finite. This defines an equivalence relation on the set of almost homomorphisms. Real numbers are defined as the equivalence classes of this relation. Alternatively, the almost homomorphisms taking only finitely many values form a subgroup, and the underlying additive group of the real number is the quotient group. To add real numbers defined this way we add the almost homomorphisms that represent them. Multiplication of real numbers corresponds to functional composition of almost homomorphisms. If denotes the real number represented by an almost homomorphism we say that if is bounded or takes an infinite number of positive values on . This defines the linear order relation on the set of real numbers constructed this way.
Faltin et al. (1975) write: "Few mathematical structures have undergone as many revisions or have been presented in as many guises as the real numbers. Every generation reexamines the reals in the light of its values and mathematical objectives."^{ [16]}
A number of other constructions have been given, by:
For an overview, see Weiss (2015).
As a reviewer of one noted: "The details are all included, but as usual they are tedious and not too instructive."^{ [17]}