# The Real Number Sequence

## References

### 2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/real_number#Definition Retrieved:2015-6-22.
• The real number system $\displaystyle{ (\mathbb R ; + ; \cdot ; \lt ) }$ 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.

### 2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Real_number#Definition
• Let R denote the set of all real numbers. Then:
• The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
• The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:
• if x ≥ y then x + z ≥ y + z;
• if x ≥ 0 and y ≥ 0 then xy ≥ 0.
• The order is Dedekind-complete; that is, every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R.
• The last property is what differentiates the reals from the rationals. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational.
• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/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.