Lambda the Ultimate

Just a small thing I've often wondered about. In math, people are often content separating algorithms into two classes: those which terminate, and those which don't. For the latter, often the bottom symbol is introduced. But isn't there obviously a third class: Those algorithms for which I don't know whether they terminate?

So, I've always wondered: Don't you introduce a logical inconsistency by cleanly separating algorithms? I.e., since you state something you can't know, it must be inconsistent, right?

Looking for views/the best answer/etc. Or, an open invitation to shoot with your best wisdom.