TITLE: A model of CH +$\lnot \diamondsuit$
QUESTION [9 upvotes]: All of the models of CH which I know of also satisfy $\diamondsuit$. What is the easiest way to produce a model of CH wherein $\diamondsuit$ is false?

REPLY [10 votes]: "The easiest way" to produce a model of CH in which $\diamondsuit$ is false is to start with a model of GCH and then do a countable support iteration of length $\omega_2$ killing off a potential $\diamondsuit$ sequence at each stage.
The forcing for doing this is straightforward: supposing $\langle A_\alpha:\alpha<\omega_1\rangle$ is a sequence with $A_\alpha\subseteq \alpha$, we force the existence of an uncountable $X\subseteq\omega_1$ such that $X\cap\alpha\neq A_\alpha$ for all $\alpha$.  This is done by viewing conditions as telling us initial segments of $X$.
Iterating the above forcing $\omega_2$ times while using bookkeeping to make sure we kill of any potential $\diamondsuit$ sequence actually works.
BUT
There's a whole lot of work involved in proving that the iteration doesn't add new reals. This is where one must wrestle with the Shelah machinery of $\mathbb{D}$-completeness and $<\omega_1$-properness