You canít define a real number such as sqrt2 without Archimedes Postulate (cuts or nests). If we assume the truth of the Pythagorean Theorem (sqrt2 exists), then Archimedes Postulate is a consequence of the Pythagorean Theorem.
But the Pythagorean theorem assumes the existence of a continuum, another neat discussion if anyone is interested.
For the time being, letís postulate the existence of a continuum. This means, for example, you can stretch a row of atoms to fit any triangle, (not provable- you can create any right angle triangle so you must have been able to stretch a row of atoms to any length. But you canít definitively prove you have a right angled triangle).
So we have:
(Archimedes Theorem iff Pythagorean Theorem) iff Continuum Hypothesis.
Yes Plato, now I am trolling. Thatís why I am posting in the DISCUSSION forum.
EDIT: Is movement continuous? But the existence of a continuum requires that you specify any point in space using only moving objects (everything moves). I think we have reached the end of the road.