I was teaching the $S_N1$ mechanism to a friend. He asked a question a subtle question I can't shake off. Take a tertiary alkyl halide and aqueous sodium hydroxide. The first step is the formation of the carbocation. My friend asked how did the alkyl halide just do that all of a sudden. What about the presence of the $\ce{NaOH}$ made this compound do this if it doesn't even interact with it ( in the first step)? I first thought maybe it was just that the first step is something that is a feasible process, and that the alkyl halide would do (to an appreciable extent) that even without the $\ce{NaOH}$, but I am not sure.
Is my idea correct?