Definition: An ordered set is order-complete if any nonempty subset with an upper bound, has a lowest upper bound or supremo.
Notation: We denote the system of first-order Peano Axioms (along with axioms for addition and multiplication) by PA1.
1.- Can we express the order-completeness of $\mathbb{N}$ using first-order logic? How does it look?
2.- Can we prove that natural numbers are order-complete using PA1 or it has to be considered as an axiom?