Is there any name for the standard ordering on the reals? I typically just see descriptors like "the usual ordering", "the typical ordering", or - well - "the standard ordering". But is there any better name for it so that we can avoid ambiguity or misinterpretation? I kinda want to call it something like "the Archimedean ordering".
Also, what is a rigorous way by which to define it. I was thinking something like this: $$(\forall x \in \mathbb{R} \,\, \& \,\, \forall y > 0), \,\, (x < x+y)$$ But, of course, this leaves one to wonder what "$y>0$" means. It also relies on the ordered set being endowed with the standard addition operator, which is not always the case and also causes us to need to define addition as well. (If you use addition or another operator in your answer, then please create a rigorous definition of what that addition or operator means too).
This has been bothering me for years, but I have only just now decided to ask about it.
The reals can be defined as the (i.e., unique up to unique isomorphism) complete ordered field. So the order is very inherent. Instead of "usual" or "standard", one might therefore refer to it as "canonical" ordering. However, I'd prefer standard. (The fact that the standard ordering has the property of being Archimedian is secondary; similarly, we wouldn't call the standard ordering of N the well-ordeing of N, but say that the standard ordering is a well-order)
We can see that the ordering of $\Bbb R$ is inherent from the fact that we can express it in terms of the field operations: $$ x\le y\iff \exists z\colon x+z^2=y.$$