Construction of real numbers: Encyclopedia II - Construction of real numbers - Synthetic approach

Construction of real numbers - Synthetic approach

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 total order ≤ on R, satisfying the following properties.

1. (R, +, *) forms a field. In other words,

• For all x, y, and z in R, x + (y + z) = (x + y) + z and x * (y * z) = (x * y) * z. (associativity of addition and multiplication)
• For all x and y in R, x + y = y + x and x * y = y * x. (commutativity of addition and multiplication)
• For all x, y, and z in R, x * (y + z) = (x * y) + (x * z). (distributivity of multiplication over addition)
• For all x in R, x + 0 = x. (existence of additive identity)
• 0 is not equal to 1, and for all x in R, x * 1 = x. (existence of multiplicative identity)
• For every x in R, there exists an element −x in R, such that x + (−x) = 0. (existence of additive inverses)
• For every x ≠ 0 in R, there exists an element x⁻¹ in R, such that x * x⁻¹ = 1. (existence of multiplicative inverses)

2. (R, ≤) forms a totally ordered set. In other words,

• For all x in R, x ≤ x. (reflexivity)
• For all x and y in R, if x ≤ y and y ≤ x, then x = y. (antisymmetry)
• For all x, y, and z in R, if x ≤ y and y ≤ z, then x ≤ z. (transitivity)
• For all x and y in R, x ≤ y or y ≤ x. (totalness)

3. The field operations + and * on R are compatible with the order ≤. In other words,

• For all x, y and z in R, if x ≤ y, then x + z ≤ y + z. (preservation of order under addition)
• For all x and y in R, if 0 ≤ x and 0 ≤ y, then 0 ≤ x * y (preservation of order under multiplication)

4. The order ≤ is complete in the following sense: every non-empty subset of R bounded above has a least upper bound. In other words,

• If A is a non-empty subset of R, and if A has an upper bound, then A has an upper bound u, such that for every upper bound v of A, u ≤ v.

When we say that any two models of the above axioms are isomorphic, we mean that for any two models (R, 0_R, 1_R, +_R, *_R, ≤_R) and (S, 0_S, 1_S, +_S, *_S, ≤_S), there is a bijection f : R → S preserving both the field operations and the order. Explicitly,

• f is both 1-1 and onto.
• f(0_R) = 0_S and f(1_R) = 1_S.
• For all x and y in R, f(x +_R y) = f(x) +_S f(y) and f(x *_R y) = f(x) *_S f(y).
• For all x and y in R, x ≤_R y if and only if f(x) ≤_S f(y).

The final axiom above is most crucial. Without this axiom, we simply have the axioms which define an ordered field, and there are many non-isomorphic models which satisfy these axioms. However, when the completeness axiom is added, it can be shown that any two models must be isomorphic, and so in this sense, there is only one complete ordered field.

Adapted from the Wikipedia article "Synthetic approach", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki