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Let $M$ be a countable transitive model of $ZFC$, $\mathbb{P}\in M$ a forcing notion and $G$ a $\mathbb{P}$-generic filter over $M$.

Assume $( \mathbb{P}$ is c.c.c and $|\mathbb{P}|\leq \omega_{1})^{M}$ and $\Diamond$ holds in $M$. I want to show $\Diamond$ is holds in $M[G]$.

How to use $ \Diamond $ in $ M $?

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  • $\begingroup$ Oops! That's the other direction! Sorry! :-) $\endgroup$
    – Asaf Karagila
    Feb 13, 2015 at 19:59
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    $\begingroup$ Apply $\diamondsuit$ to the $\mathbb{P}$-names for subsets of $\omega_1$? (By the ccc, you can code such a name as an $\omega_1$-sequence of countable subsets of $\mathbb{P}$.) $\endgroup$ Feb 13, 2015 at 22:49

1 Answer 1

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Here are the details on Paul's comments (just copy pasting from my tex files - Replace $\kappa$ and $S$ by $\omega_1$ below):

For a regular uncountable cardinal $\kappa$ and a stationary $S \subseteq \kappa$,

$\diamondsuit_S$ is the statement: There exists a sequence $\langle A_{\alpha} : \alpha \in S \rangle$ such that each $A_{\alpha} \subseteq \alpha$ and for every $A \subseteq \kappa$, the set $\{\alpha \in S : A_{\alpha} = A \cap \alpha\}$ is stationary in $\kappa$.

Claim: Suppose $\kappa$ is a regular uncountable cardinal and $S$ is a stationary subset of $\kappa$. Let $P$ be a $\kappa$-c.c. poset of size $\leq \kappa$. Assume $\diamondsuit_S$ holds in $V$. Let $G$ be $P$-generic over $V$. Then $V[G] \models \diamondsuit_S$

[Why? Using $|P| = \kappa$, nice $P$-names for subsets of $\kappa$ (union of sets of form $W \times \{\alpha\}$ where $\alpha < \kappa$ and $W$ is an antichain in $P$) can be thought of as subsets of $\kappa$ by identifying $P \times \kappa$ with $\kappa$. If $A \subseteq \kappa$ codes such a nice name, then let $n(A)$ denote the corresponding $P$-name. Fix a $\diamondsuit_S$ witnessing sequence $\langle A_{\alpha} : \alpha \in S \rangle$ in $V$. In $V[G]$, define $\langle B_{\alpha} : \alpha \in S \rangle$ by $B_{\alpha} = \text{eval}_G(n(A_{\alpha})) \cap \alpha$ if $A_{\alpha}$ codes a nice name for a subset of $\kappa$, otherwise $B_{\alpha} = 0$. Fix a code $A$ for a nice $P$-name $n(A)$ for a subset of $\kappa$. Then for each $\alpha < \kappa$, $A \cap \alpha$ also codes a nice name. Moreover, using $\kappa$-ccness of $P$, the set $\{\alpha < \kappa : \text{eval}_G(n(A \cap \alpha)) \cap \alpha = \text{eval}_G(n(A)) \cap \alpha \}$ contain a club. Since $A \cap \alpha = A_{\alpha}$ on a stationary subset of $S$, $B_{\alpha} = \text{eval}_G(n(A)) \cap \alpha$ on this stationary subset as well. Hence $\langle B_{\alpha} : \alpha \in S \rangle$ witnesses $\diamondsuit_S$ in $V[G]$.]

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