A model of CH +$\lnot \diamondsuit$

Question

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?

Answer 1

"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

Answer 2

The first one of these was probably Jensen's model for the consistency of Suslin's Hypothesis (SH) with the Continuum Hypothesis (CH). This is by no means an easy model, e.g. it involves an iteration of length $\omega_2$ which is neither finite support nor countable support. Devlin and Johnsbråten wrote a monograph about it in the 70's titled The Souslin Problem (LNM 405).

There are better ways to do this nowadays. For example, Shelah has developed a variety of new forcings to kill Suslin trees and not add reals. These can be used to force CH + SH using a regular countable support iteration. In fact, because of advances in technology for preserving 'no new reals' there are probably lots more models of CH + $\lnot\diamondsuit$ out there.

As for general principles, the P-ideal dichotomy is compatible with CH but incompatible with $\diamondsuit$. There are probably lots more, but I'd have to check.

On the other hand, it is well known that it is very easy to inadvertently force $\diamondsuit$. For example, Rosłanowski and Shelah have shown that any proper forcing of size at most $2^{\aleph_0}$ that collapses $\aleph_2$ must force $\diamondsuit$ (and hence must collapse $2^{\aleph_0}$ too). Therefore, the only reliable way to force CH + $\lnot\diamondsuit$ via a countable support iteration proper forcing is to start with a model of CH and not add new reals.

Answer 3

Todd Eisworth and Justin Moore gave a series of lectures on this precise topic and are preparing a monograph based on their lectures. Preliminary notes are available I believe.