TITLE: How hard is it to destroy a diamond? (with a real)
QUESTION [28 upvotes]: 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:

Blowing up the continuum,
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)?

REPLY [10 votes]: 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.
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