Talk:Negative Infinity is Minimal

Lord_Farin, I don't know enough to make the duality explicit. If you want, I can just copy the proof and substitute as needed, but that seems a waste of time. --Dfeuer (talk) 23:00, 6 January 2013 (UTC)

Since I don't see immediately how to apply duality here (except that it's intuitively clear) I think it's best to do precisely that - even though it seems a waste of time. --Lord_Farin (talk) 08:23, 7 January 2013 (UTC)

Once we figure out how the proof should go over at Positive Infinity is Maximal, I guess I can plod through it. Unless you have some brilliant categorical insight before then. --Dfeuer (talk) 08:25, 7 January 2013 (UTC)

If you want to use it to prove $\le$ is an ordering (which seems convenient), we cannot use things about order duality, I fear... So that insight won't come. --Lord_Farin (talk) 08:38, 7 January 2013 (UTC)

Like I said, I don't know much about such things, but there's a symmetry inherent to the situation that should be amenable to some kind of formal analysis... --Dfeuer (talk) 08:48, 7 January 2013 (UTC)

I think that what you are looking for is that for $x,y \in \overline \R$, $x \le y$ iff $-y \le -x$ (although $-(+\infty)$ is not defined, since $(-\infty) + (+\infty)$ is not defined). It's not easily made formal at this point, but I have some ideas for more general results sprouting from this observation. --Lord_Farin (talk) 10:11, 7 January 2013 (UTC)