Good Math

The Reals, Axiomatically

The axiomatic definition is, in many ways, quite similar to the definition the number line provides, but it does the job in a very formal way. An axiomatic definition doesn’t tell you how to get the real numbers; it just describes them with rules that draw on simple set theory and logic.

When we’re building something like the real numbers, which are defined by a set of related components, mathematicians like to be able to say that what we’re defining is a single object. So we define it as a tuple. There’s no deep meaning to the construction of a tuple; it’s just a way of gathering components into a single object.

The reals are defined by a tuple: (R, +, 0, ×, 1, ≤), where R is an infinite set, “+” and “×” are binary ...