Archimedean field

This article defines a field property: a property that can be evaluated to true/false for any field.
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Definition

An Archimedean field is a field with a total ordering $\leq$ on its elements satisfying the following:

$0\leq 1$ .
$a\leq b\implies -b\leq -a$ .
$a\leq b$ and $c\leq d$ implies that $a+c\leq b+d$ .
$0\leq a$ and $b\leq c$ implies that $ab\leq ac$ .
The Archimedean property: For any $x\in K$ , there exists a natural number $n$ such that $x\leq n$ , where the natural number $n$ is viewed as the element of $K$ obtained by adding $1$ to itself $n$ times.