It is well-known that the continuum hypothesis (CH) holds in the constructible universe $L$. Now suppose we add a nonconstructible subset $a_1$ of $\omega$, and consider the universe $L[a_1]$. Will CH be automatically true or automatically false in this new model, or is there an independence result?
If $CH$ still holds in $L[a_1]$, we may still add a subset $a_2$ of $\omega$ with $a_2 \not\in L[a_1]$. As long as CH is false, we may add a new subset of $\omega$ (in fact we can add nonconstructible elements as we please, regardless of whether CH holds or not). So for each ordinal $\alpha$, we have an axiom ${\sf Ax}_{\alpha}$ stating the existence of a function $a:\alpha \to {\cal P}(\omega)$ such that $a(\beta) \not\in L[(a(\gamma)_{\gamma < \beta})]$ for all $\beta < \alpha$.
The question can then be stated as, is there an $\alpha$ such that $ZFC+{\sf Ax}_{\alpha}$ implies that CH is false ? What is the smallest such $\alpha$ ?