If we start with $V\models\lozenge$, it is not hard to force the failure of diamond. You can blow up the continuum, or destroy all the Suslin trees. You can blow up the continuum of $\aleph\_1$, and then collapse $\aleph\_1$ to be countable.


There are many ways of doing that, but all of them (that I could think of, with the help of a few people over the day) include one of the two:


1. Blowing up the continuum,
2. Collapsing cardinals.



> 
> Is it consistent that $V\models\lozenge$, and $r$ is a $V$-generic real such that $V[r]\models\lnot\lozenge+\sf CH$ and no cardinals were collapsed between $V$ and $V[r]$?
> 
> 
> 


If the answer is positive, can we strengthen the preservation of $\sf CH$ by requiring also that the continuum function remains the same (so no blowing up power sets of larger cardinals somehow)?



How hard is it to destroy a diamond? (with a real)
The answer is yes, assuming the existence of $\aleph\_2$-many measurable cardinals. To see this, assume $GCH+\Diamond$ holds and $S$ is a discrete set of measurable cardinals of size $\aleph\_2.$


**Step 1.** Force with Prikry product forcing $P\_S$ to change the cofinality of each element of $S$ into $\omega.$


Note that the extension is of the form $V[(x\_\alpha: \alpha\in S)]$, where each $x\_\alpha$ is an $\omega-$sequence cofinal in $\alpha.$


**Step 2.** Force with Jensen's coding theorem, to code everything into a real $r$, so that we have $V[(x\_\alpha: \alpha\in S)][r]=V[r]$ (we can do this using a set forcing construction, and assuming that the ground model is a core model).


**Step 3.** Force over $V[r]$, by a cardinal and $GCH$ preserving forcing iteration to force $\neg \Diamond$.


Note that the generic can be seen as a subset $X$ of $S.$ Now working in $V[r][X],$
define a new sequence $(y\_\alpha: \alpha\in S),$ so that $y\_\alpha=x\_\alpha,$ if $\alpha\in X$ and $y\_\alpha=x\_\alpha\setminus\{ min(x\_\alpha) \}$ if $\alpha\notin X.$


Then let $V\_1=V[(y\_\alpha: \alpha\in S)]$ and $V\_2=V\_1[r].$ Note that:


1) $V\_1\models GCH+\Diamond,$


2) $V\_2=V[(y\_\alpha: \alpha\in S)][r]=V[r][X]\models GCH+\neg\Diamond,$


3) $V\_2=V\_1[r]$, for some real $r$.


I may mention that the above method can be used to prove the consistency of many statements using adding a single real.


