Existence of the supremum can't be proven using field and order axioms

I'm reading through "Introduction to Real Analysis" by Robert G. Bartle and Donald R. Sherbert, fourth edition, page 39.

But how does he know that the existence of the supremum can't be proven using the field and order axioms? is there a way to show that the supremum is not a consequence of those axioms? I'm thinking of trying but I can't see the end of it, that is to say, play around with the field and order axioms (and theorems derived from them) to see if it's possible to proof the existence of the supremum, but the fact that I can't doesn't mean it can't be done.

Thank you