Page "Axiom" Paragraph 15

The object of study is the real numbers.

The real numbers are uniquely picked out ( up to isomorphism ) by the properties of a Dedekind complete ordered field, meaning that any nonempty set of real numbers with an upper bound has a least upper bound.

However, expressing these properties as axioms requires use of second-order logic.

The Löwenheim-Skolem theorems tell us that if we restrict ourselves to first-order logic, any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger.

Some of the latter are studied in non-standard analysis.