The Order Axioms of ℝ Notes

The real numbers are more than a field—they are an ordered field.

Definition: A set is ordered if there exists a binary relation "<" satisfying:

Axioms of Order:

1. Trichotomy: For all

   $a, b \in ℝ$

   a,b∈R, exactly one of the following is true:

   • $a < b$

     a<b

   • $a = b$

     a=b

   • $a > b$

     a>b

2. Transitivity: If

   $a < b$

   a<b and

   $b < c$

   b<c, then

   $a < c$

   a<c

3. Addition Property: If

   $a < b$

   a<b, then

   $a + c < b + c$

   a+c<b+c

4. Multiplication Property: If

   $0 < a$

   0<a and

   $0 < b$

   0<b, then

   $0 < ab$

   0<ab

From these, we define:

• $a ≤ b$

  a≤b means

  $a < b$

  a<b or

  $a = b$

  a=b

• $a ≥ b$

  a≥b means

  $b ≤ a$

  b≤a