The synthetic approach axiomatically defines the real number system as a complete ordered field. Precisely, this means the following. A model for the real number system consists of a set R, two distinct elements 0 and 1 of R, two binary operations + and * on R (called addition and multiplication, resp.), a binary relation ≤ on R, satisfying the following properties.
1. (R, +, *) forms a field. In other words,
The final axiom, defining the order as Dedekind-complete, is most crucial. Without this axiom, we simply have the axioms which define a totally ordered field, and there are many non-isomorphic models which satisfy these axioms. This axiom implies that the Archimedean property applies for this field. Therefore, when the completeness axiom is added, it can be proved that any two models must be isomorphic, and so in this sense, there is only one complete ordered Archimedean field.
When we say that any two models of the above axioms are isomorphic, we mean that for any two models (R, 0R, 1R, +R, *R, ≤R) and (S, 0S, 1S, +S, *S, ≤S), there is a bijection f : R → S preserving both the field operations and the order. Explicitly,
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.
If we have a space where Cauchy sequences are meaningful (such as a metric space, i.e., a space where distance is defined, or more generally a uniform space), a standard procedure to force all Cauchy sequences to converge is adding new points to the space (a process called completion). By starting with rational numbers and the metric d(x,y) = |x − y|, we can construct the real numbers, as will be detailed below. (A different metric on the rationals could result in the p-adic numbers instead.)
Let R be the set of Cauchy sequences of rational numbers. Cauchy sequences (x) and (y) can be added, multiplied and compared as follows:
Two Cauchy sequences are called equivalent if the sequence (xn - yn) has limit 0. This does indeed define an equivalence relation, it is compatible with the operations defined above, and the set R of all equivalence classes can be shown to satisfy all the usual axioms of the real numbers. We can embed the rational numbers into the reals by identifying the rational number r with the equivalence class of the sequence (r,r,r, …).
The only real number axiom that does not follow easily from the definitions is the completeness of ≤. 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, there is a rational number L such that L < s for some s in S. Now define sequences of rationals (un) and (ln) as follows:
For each n consider the number:
If mn is an upper bound for S set:
This obviously defines two Cauchy sequences of rationals, and so we have real numbers l = (ln) and u = (un). It is easy to prove, by induction on n that:
Thus u is an upper bound for S. To see that it is a least upper bound, notice that the limit of (un − ln) is 0, and so l = u. Now suppose b < u = l. Since (ln) is monotonic increasing it is easy to see that b < ln for some n. But ln is not an upper bound for S and so neither is b. Hence u is a least upper bound for S and ≤ is complete.
A practical and concrete representative for an equivalence class representing a real number is provided by the representation to base b -- in practice, b is usually 2 (binary), 8 (octal), 10 (decimal) or 16 (hexadecimal). For example, the number π = 3.14159... corresponds to the Cauchy sequence (3,3.1,3.14,3.141,3.1415,...). Notice that the sequence (0,0.9,0.99,0.999,0.9999,...) is equivalent to the sequence (1,1.0,1.00,1.000,1.0000,...); this shows that 0.999... = 1.
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 fulfils the following conditions:
Leftrightarrow x subset y
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 . It can be seen from the definitions above that is a real number, and that .
We can take the infinite decimal expansion to be the definition of a real number, considering expansions like 0.9999... and 1.0000... to be equivalent, and define the arithmetical operations formally. Similarly another radix can be used. This is a special case of the construction by Cauchy sequences.
As in the hyperreal numbers, one constructs the hyperrationals *Q from the rational numbers by means of an ultrafilter. Here a hyperrational is by definition a ratio of two hyperintegers. Consider the ring B of all bounded (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. Note that B is not an internal set in *Q.
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.
A relatively less known construction allows to define real numbers using only the additive group of integers. Different versions of this construction are described in , and The construction has been formally verified by the IsarMathLib project
Let an almost homomorphism be a map such that the set is finite. 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. To add real numbers defined this way we add the almost homomorphisms that represent them. Multiplication of real numbers corresponds to 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.