Completeness Axiom

The field and order axioms for \mathbb{R} and various other concepts connected with these as given enable us to make algebraic computations with real numbers involving a finite number of operations of addition, multiplication, subtraction and division. These axioms and concept are independently satisfied by the set of rational numbers \mathbb{Q}. As such, these are sufficient to make distinction between the sets \mathbb{Q} and \mathbb{R}.

An additional axiom C, known as completeness axiom, distinguishes \mathbb{Q} from \mathbb{R} and has important consequences. This axiom confirms the existence of the unique supremum and the infimum of sets according as they are bounded above or below. It is only due to this axiom that the existence of irrational numbers could be justified.

Completeness Axiom in \mathbb{R}. Every non-empty set of real numbers which is bounded above has a supremum in \mathbb{R}. In other words, the set of upper bounds of a non-empty set bounded above has a least member. This axiom is also known as the continuity axiom in \mathbb{R}. If S is a set bounded below, then by considering the set T = \left\{ {x: - x \in  S} \right\} we shall state the completeness axiom in the alternate form as:

Every non-empty set of real numbers which is bounded below has as infimum in \mathbb{R}.