Axioms
In order to define the real numbers, we need a set of axioms that the real numbers will obey.
In order to define the real numbers, we need a set of axioms that the real numbers will obey.
The real numbers are a set denoted by \mathbb{R}, such that the following properties hold
(R1) \mathbb{R} is a field under addition and multiplication.
(R2) \mathbb{R} is totally ordered for some \leq.
(R3) Addition and multiplication preserve order.
(R4) Every non-empty subset of \mathbb{R} that is bounded above has a least upper bound.