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?



A model of CH +$\lnot \diamondsuit$
"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


