In mathematics, there are a number of 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 ...

Including:

• Construction of real numbers - Synthetic approach
• Construction of real numbers - Explicit constructions of models
• Construction of real numbers - Construction from Cauchy sequences
• Construction of real numbers - Construction by Dedekind cuts
• Construction of real numbers - Construction by decimal expansions
• Construction of real numbers - Construction from ultrafilters
• Construction of real numbers - Construction from surreal numbers
• Construction of real numbers - Construction from the group of integers

Read more here: » Construction of real numbers: Encyclopedia - Construction of real numbers

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, resp.), a total order ≤ on R, satisfying the following properties. 1. (R, +, *) forms a field. In other words, For all x, y, and z in RSee also:

Construction of real numbers, Construction of real numbers - Synthetic approach, Construction of real numbers - Explicit constructions of models, Construction of real numbers - Construction from Cauchy sequences, Construction of real numbers - Construction by Dedekind cuts, Construction of real numbers - Construction by decimal expansions, Construction of real numbers - Construction from ultrafilters, Construction of real numbers - Construction from surreal numbers, Construction of real numbers - Construction from the group of integers

Read more here: » Construction of real numbers: Encyclopedia II - Construction of real numbers - Synthetic approach