On the least upper bound property
On the least upper bound property [ edit ] Axiom 4, which requires the order to be Dedekind-complete , implies the Archimedean property . The axiom is crucial in the characterization of the reals. For example, the totally ordered field of the rational numbers Q satisfy the first three axioms, but not the fourth. In other words, models of the rational numbers are also models of the first three axioms. Note that the axiom is not firstorderizable , as it expresses a statement about collections of reals and not just individual such numbers. As such, the reals are not given by a first-order logic theory . On models [ edit ] A model of real numbers is a mathematical structure that satisfies the above axioms. Several models are given below . Any two models are isomorphic; so, the real numbers are unique up to isomorphisms. Saying that any two models are isomorphic means that for any two models...