♪ (J. Stewart. Calculus 6th ed. pp 682) Axiom of Completeness = AoC
= A nonempty set of real numbers that has an upper bound has a least upper bound.
AoC is an expression of the fact that there is no gap or hole in the real number line.
1. Intuition please on the Axiom of Completeness? How does AoC warrant 'no gap or hole'?
2. (http://www.physicsforums.com/showthread.php?t=648536) Person micromass says this expression is 'rubbish' and then 'good intuition'? I'm nonplussed. What's flawed and immaculate?
♪ pp 4 of 7 (http://www.maths.qmul.ac.uk/~klages/mth4100/week1.pdf) In summary, the real numbers R are complete in the sense that they correspond to all points on the real line, i.e., there are no “holes” or “gaps”, whereas the rationals have ”holes” (namely the irrationals).
♪ pp 1 of 8 (ttp://www.nku.edu/~longa/classes/2011fall/mat420/days/highlights/highlights1.3.pdf)
There are “holes” in the set of rational numbers – interesting numbers that aren’t rational. This completeness axiom merely asserts that all the holes are filled in for the real numbers (with the additional of the irrational numbers).