Hi zli034,
May I ask why you need this? Maybe my explanation will be better if I know this.
Basically, we want to generalize number system such as \mathbb{R} and \mathbb{Q}. In particular, there are some results on these systems that can be explained by means of a minimal number of axioms. Why do we do this? First, to get a better understanding of \mathbb{R} and \mathbb{Q}. Second, because there are a lot of other systems out there which also share the same properties and the theory of ordered fields will unify these systems.
What do we have on \mathbb{R} and \mathbb{Q}? Well, we have an addition. This addition + satisfies following properties:
- Associativity: a+(b+c)=(a+b)+c
- Neutral element: a+0=a=0+a
- Inverse element: a+(-a)=0=(-a)+a
- Commutativity: a+b=b+a
Anything else which shares the same properties is called an abelian group. The theory of abelian groups is extremely useful and it arises everywhere!
Now, what else do we have on \mathbb{R} and \mathbb{Q}? A multiplication of course? This satisfies:
- Associativity: a(bc)=(ab)c
- Neutral element: a1=a=1a
- Inverse element: aa^{-1}=1=a^{-1}a for all nonzero a.
- Commutativity: ab=ba
- Distributivity: a(b+c)=ab+ac
This is called a field. Other fields include the complex numbers. But there is something on \mathbb{R} and \mathbb{Q} that the complex numbers don't have: an order!
Basically, we have a relation \leq that satisfies
- Reflexivity: a\leq a
- Anti-symmetry: a\leq b~\text{and}~b\leq a implies a=b
- Transitivity: a\leq b~\text{and}~b\leq c implies a\leq c
This is what we call a partial order. But there order on \mathbb{R} and \mathbb{Q} satisfies some additional properties:
- Every two elements are comparable: for all a,b we have either a\leq b or b\leq a (or both).
- If a\leq b, then a+c\leq b+c
- If 0\leq a and 0\leq b, then 0\leq ab
A structure which satisfies all these axioms is called an ordered field...
