The empty set $\emptyset$ here is the forcing condition that has the least amount of information about the generic object being constructed.

Since it has the least information, you might think it should be at the bottom of the partial order. There are two replies to this:

(1) Mathematically, we are free to define our partial order however we want, even if it confuses the reader.

(2) In this case, there is no intent to confuse the reader. Having the *least* information, the condition $\emptyset$ also leaves the *greatest* amount of possibilities still open. So it is really a 50-50 decision which way one should define it. (Others may know the history of this particular convention.)

A similar convention question is whether the Turing degree of the computable sets, $\mathbf 0$, is the smallest or the largest Turing degree. Conventionally one speaks of the degrees of *unsolvability* and so $\mathbf 0$ is the smallest, but if the convention were degrees of *solvability* it would be the greatest. Since none of the other degrees are very solvable at all, the current convention is perhaps the best in this case.

anyuniverse, whether or not this ground universe satisfies CH. (Of course, if your ground model violates CH, then you don'tneedto force over it at all.) – Andreas Blass Mar 10 '12 at 17:21