The Axiom of Completeness states that if a non-empty set is bounded above and has at least one element, then the set contains a least upper bound. The Axiom of Completeness is what separates the from .

After reading about the rational numbers and realizing that although has an infinite number of values, it actually has more gaps than values. The Axiom of Completeness eliminates all gaps in the real number line and assures us by assumption that any and all real values exist.

Ex. Suppose the set A = { x : x² < 2 }. One upper bound for this set is 2. Another is 1.4143. Another is 1.41421357. Dealing strictly in the number set for a moment, notice that we can get arbitrarily close to the sqrt(2), but we can never precisely describe it within . Now suppose we are back within . We have a value, precisely the sqrt(2), which is not in the set A, but is still the least upper bound for A. We know this because if we pick any arbitrarily small 1 > > 0, the value (sqrt(2) - ) exists within A. We know that (sqrt(2) - ) is not the least upper bound because (sqrt(2) - /2) is also in A.