Construction of the real numbers
[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.
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
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.
For example, the totally ordered field of the rational numbers Q satisfies the first three axioms, but not the fourth.
Note that the axiom is nonfirstorderizable, as it expresses a statement about collections of reals and not just individual such numbers.
A model of real numbers is a mathematical structure that satisfies the above axioms.
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
is a linearly ordered abelian group under addition with distinguished element 1.
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.
Cauchy sequences (xn) and (yn) can be added and multiplied as follows: Two Cauchy sequences (xn) and (yn) are called equivalent if and only if the difference between them tends to zero; that is, for every rational number ε > 0, there exists an integer N such that for all natural numbers n > N, one has |xn − yn| < ε.
by identifying a rational number r with the equivalence class of the Cauchy sequence (r, r, r, ...).
This reflects the observation that one can often use different sequences to approximate the same real number.
[6] The only real number axiom that does not follow easily from the definitions is the completeness of ≤, i.e. the least upper bound property.
This defines two Cauchy sequences of rationals, and so the real numbers l = (ln) and u = (un).
The usual decimal notation can be translated to Cauchy sequences in a natural way.
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
[12] This construction uses a non-principal ultrafilter over the set of natural numbers, the existence of which is guaranteed by the axiom of choice.
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
[13][14][15] Arthan (2004), who attributes this construction to unpublished work by Stephen Schanuel, refers to this construction as the Eudoxus reals, naming them after ancient Greek astronomer and mathematician Eudoxus of Cnidus.
As noted by Shenitzer (1987) and Arthan (2004), Eudoxus's treatment of quantity using the behavior of proportions became the basis for this construction.
This construction has been formally verified to give a Dedekind-complete ordered field by the IsarMathLib project.
Multiplication of real numbers corresponds to functional composition of almost homomorphisms.
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.
