If i have a set of numbers, that contains every positive number, including 0 , except the negative ones, can i claim it as an ordered field? I would say no, because you can't find an $y$ for any $x$ so that: $$x+y=0$$
So one Axiom for fields is violated. I am not quite sure, is this conclusion correct? Are there any other axioms for ordered fields violated? I can just think of this one. Thanks for any hints.