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.