Explain the main ideas behind the construction of the real n

Explain the main ideas behind the construction of the real numbers as the complete ordered field. Why is ordered-field completeness an important mathematical concept?

Solution

This first set of axioms are called the field axioms because any object satisfying them is called a field.

Axiom 1 (Associative Laws). If a, b, c F, then (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).

Axiom 2 (Commutative Laws). If a, b F, then a + b = b + a and a × b = b × a.

Axiom 3 (Distributive Law). If a, b, c F, then a × (b + c) = a × b + a × c.

Axiom 4 (Existence of identities). There are 0, 1 F such that a +0= a and a × 1 = a, a F.

Axiom 5 (Existence of an additive inverse). For each a F there is a F such that a + (a) = 0.

Axiom 6 (Existence of a multiplicative inverse). For each a F \\ {0} there is a1 F such that a × a1 = 1.

Every set which is bounded above has a least upper bound. This is the final axiom. Any set which satisfies all eight axioms is called a complete ordered field. We assume the existence of a complete ordered field, called the real numbers