Because the nonnegative rationals $\mathbb{Q}_+$ are countable, I can enumerate them as a sequence $\{q_i\}_{i\in\mathbb{N}}$. For any pair of rationals $q$ and $p$, I can perform the comparison $q\leq p$ according to the ordering of the real numbers. Is it possible to enumerate the rationals as an increasing sequence, i.e., $q_i\leq q_{i+1}$ for all $i\in\mathbb{N}$? More generally, given an ordering on $\mathbb{Q}_+$, can I enumerate $\mathbb{Q}_+$ as an increasing sequence under this ordering?
I do not think this is true, but I cannot think of a good approach to proving it false.
EDIT: I meant to only ask about the nonnegative rationals, so I am correcting my question to fix that mistake. The original version of the question asked about enumerating the rationals.