Suppose we define the real numbers as an ordered field satisfying the least upper bound property. Is there only one set $\mathbf E$ that satisfies the axioms listed below?
- For all $x\in\mathbb R$, exactly one of the following holds: $x=0$, or $x\in \mathbf E$, or $-x\in\mathbf E$.
- If $a\in\mathbf E$ and $b\in\mathbf E$, then $a + b\in\mathbf E$.
- If $a\in\mathbf E$ and $b\in\mathbf E$, then $a \cdot b\in\mathbf E$.
I'm interested in defining the real numbers axiomatically. It is common to state the properties of the positive real numbers as above, but I have never seen a proof that the above axioms uniquely characterise the positive real numbers. (Perhaps this is because they don't. If this is the case, then I would be interested in learning about an additional property that would uniquely characterise the set of positive reals.)
Best Answer
For $x>0$ define $\sqrt{x}$ to be the least upper bound of the set $\{y|y^2\leq x\}$.
Suppose we have a set $E$ satisfying your axioms.
If $x>0$ then $$x=(\sqrt{x})(\sqrt{x})=(-\sqrt{x})(-\sqrt{x}).$$
Either $ \sqrt{x}\in E$ or $-\sqrt{x}\in E$ so either way $x\in E$.
Conversely if $x<0$ then $-x\in E$ so $x\notin E$.
Thus $E$ is just the positive reals.