I've been reading about ordered fields and completely ordered fields, and I'm stuck on the difference between the rational and real numbers that makes the rational numbers an ordered field and the real numbers a completely ordered field. I see that the reals are complete, and the rationals aren't, but how does that correlate with one being a completely ordered field and the other just an ordered field?
My book, "A Friendly Introduction to Analysis" by Witold A. J. Kosmala, gives the following so-called 'completeness axiom' that separates a completely ordered field from an ordered one:
I understand what the axiom says, but I have a hard time grasping what it really means for the field that it applies to, and what it really means for the difference between a completely ordered and an ordered field. Furthermore, why do the rationals not satisfy this axiom? My thoughts were that a subset of the rationals also contain a supremum, namely the greatest rational number in that subset. What part don't I understand?
All help is appreciated.