I want to think of ZFC as not fully determining the powerset of the naturals, because you can add subsets with forcing and otherwise have a lot of control over the cardinality of the powerset of the naturals. But that suggests the question: what subsets does ZFC determine? There is also the related question: how complicated are the sets added by forcing?


Since Turing machines are absolute, models of ZFC should have the same hyperarithmetical sets. So $\Delta\_1^1$ is absolute. 


Can Cohen forcing add a $\Pi\_1^1$ set? If not, what is the descriptive complexity of sets added by Cohen forcing?


This isn't my field so I apologize if this question is ill formed or naive. References would also be appreciated.


Edit: I'm specifically interested in where a set/real $c$ added by forcing fits into the lightface hierarchy.



What is the descriptive complexity of a set added by Cohen forcing?
Since Cohen forcing is weakly homogeneous, all hereditarily ordinal definable sets in the Cohen extension are already in the ground model. That applies in particular to any ordinal definable real, and (in even more particular) to any real with a lightface $\Sigma^m\_n$ definition.


