The Completeness Axiom
) states that, if a set
of Real Numbers
has an upper bound, then it has a least upper bound
, known as a supremum
. Similarly it has a greatest lower bound
, known as an infimum
, if it has a lower bound.
(A set N is bounded above if there exists a number M such that x ≤ M for all x ∈ N. N is bounded below if there exists M such that x ≥ M for all x ∈ N. A set is "bounded" (bounded above and below) if there exists a number M such that |x| ≤ M for all x ∈ N.)
Note that the supremum (or infimum) need not be in the set. For instance, if you have the set of all real numbers between 0 and 1, inclusive, 1 is the supremum and in the set. However, if you have all real numbers greater than zero but less than 1, 1 is the supremum and not in the set.
My math prof at The D explained that it's because of the Completeness Axiom that the Greeks didn't invent Calculus. They were smart like Newton and Leibniz and their pals, but they only had rational numbers in their system. So if you've got the set of all x2 < 2 (for all x rational), this won't have a rational least upper bound, since the least upper bound, as we know it, is the square root of 2, which is, of course, an irrational number. The Completeness Axiom fails, things fall apart, Ack!
/me has found a way to study and write factual nodes at the same time...this is a good thing.
With special thanks to HTML Symbol Reference