I am reading a paper by Richard Weimer called "Can the complex numbers be ordered?" and he makes the following claim.
Let $G^+=\{a+bi : a,b$ are positive integers $ \}$ and let $<$ denote lexicographical ordering. It is easily demonstrated that there are an infinite number of Gaussian integers with respect to $<$ between $(a,b)$ and $(a+1,b)$ where $a,b$ are positive integers. (I understand this sentence). Thus, one can easily deduce that $G^+$ under $<$ is not well ordered, i.e., not every non-empty subset of $G^+$ possesses a smallest element. (This sentence I don't understand.)
If you have a nonempty subset $S$ of $G^+$, can't you find a minimum $c$ of the real components of elements in $S$ since $a$ is a positive integer? Then if you look at the subset $R = \{s \in S : \operatorname{Re}(s) = c\}$, won't it have a least element which will also be the least element of $S$?