Why does forcing seem to be so vacuously true?


It seems like you are just reversing the subset containment of the model of ZFC + CH to be the other way in the poset. So, why is this valid? Why are you allowed to just put the empty set at the top of the partial order\*?


\*I have just started learning set theory so I have only seen one example of forcing and it was the forcing of the set of reals to be equal to the second uncountable cardinal.



The consistency of ZFC + CH gives the ability to travel to a universe which models ZFC + \neg CH?
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.


