In 1 Operations we begin axiomatizing the real numbers by axiomatizing their operations of addition and multiplication, leading to the field axioms.
In 2 Order we define the notion of inequality in terms of the notion of positivity which we axiomatize, leading to the definition of an ordered field.
In 3 Completeness we look to formalize the notion of limit used by the babylonians and archimedes, and end up with the Nested Interval Property. This leads us to introduce new concepts (infima and suprema) and a new axiom: completeness.
In 4 The Real Numbers we define the real numbers as the (unique) complete, ordered field and study its properties.
In ?sec-numbers-functions we look at the modern defintion of real valued functions, and some of the monstrous objects this allows.