Real Numbers: Axioms or Theorem?

[LikeMath]

Hi there,

In most books that I saw, the set of real numbers under the usual sum and product is considered as a Field and say that's by the field axioms. But I have surprised when I have seen, it is a theorem. The question, are these axioms? or can they be proved?

Thank you very much.

 

[HallsofIvy]

What are "axioms" and what are "theorems" depends upon what level you are working at.

We define a field to be a set of objects, S, together with two binary operations, + and *, satisfying the "field axioms".

We can the prove, as a theorem, that a particular set, with given binary operations, such as the rational numbers or real numbers, satisfy those axioms.

 

[LikeMath]

Thank you