VTRMC 1980 Problem 7

Let $S$ be the set of all ordered pairs of integers $(m,n)$ satisfying $m>0$ and $n<0.$ Let $<$ be a partial ordering on $S$ defined by the statement $(m,n)<(m',n')$ if and only if $m\le m'$ and $n\le n'.$ An example is $(5,-10)<(8,-2).$ Now let $O$ be a completely ordered subset of $S,$ in other words if $(a,b)\in O$ and $(c,d) \in O,$ then $(a,b)<(c,d)$ or $(c,d)<(a,b).$ Also let $O'$ denote the collection of all such completely ordered sets.

(a) Determine whether and arbitrary $O\in O'$ is finite. (b) Determine whether the carnality $|O|$ of $O$ is bounded for $O\in O'.$ (c) Determine whether $|O|$ can be countable infinite for any $O\in O'.$