# Real Number

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## References

### 2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/real_number#Definition Retrieved:2015-6-22.
• The real number system $(\mathbb R ; + ; \cdot ; \lt )$ can be defined axiomatically up to an isomorphism, which is described hereafter. There are also many ways to construct "the" real number system, for example, starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers. Another possibility is to start from some rigorous axiomatization of Euclidean geometry (Hilbert, Tarski, etc.) and then define the real number system geometrically. From the structuralist point of view all these constructions are on equal footing.

### 2000

1. More precisely, given two complete totally ordered fields, there is a unique isomorphism between them. This implies that the identity is the unique field automorphism of the reals that is compatible with the ordering.