Construction of the real numbers
In mathematics, there are several ways of defining the real number system as an ordered field. The synthetic approach gives a list of axioms for the real numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are isomorphic. Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered field.
The synthetic approach axiomatically defines the real number system as a complete ordered field. Precisely, this means the following. A model for the real number system consists of a set R, two distinct elements 0 and 1 of R, two binary operations + and × on R (called addition and multiplication, respectively), and a binary relation ≤ on R, satisfying the following properties.
The rational numbers Q satisfy the first three axioms (i.e. Q is totally ordered field) but Q does not satisfy axiom 4. So axiom 4, which requires the order to be Dedekind-complete, is crucial. Axiom 4 implies the Archimedean property. Several models for axioms 1-4 are given below. Any two models for axioms 1-4 are isomorphic, and so up to isomorphism, there is only one complete ordered Archimedean field.
When we say that any two models of the above axioms are isomorphic, we mean that for any two models (R, 0R, 1R, +R, ×R, ≤R) and (S, 0S, 1S, +S, ×S, ≤S), there is a bijection f : R → S preserving both the field operations and the order. Explicitly,
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