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Let $\kappa$ be supercompact. Then the (supercompact) Laver diamond holds at $\kappa$: There is $f:\kappa\to V_\kappa$ such that for all $\lambda\geq \kappa$ and $x\in H(\lambda^+)$ there is $j:V\to M$ witnessing the $\lambda$-supercompactness of $\kappa$ and $j(f)(\kappa)=x$. This implies the usual diamond on $\kappa$.

Suppose $P$ is a $\kappa$-c.c. poset with $|P|\leq \kappa$. What can we say about the diamonds that hold at $\kappa$ in the generic extension?

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  • $\begingroup$ Isn't diamond still true? Use the ground model diamond to define $\dot{X}_\alpha$ where $\dot{X}_\alpha$ is a P-name of a subset of $\alpha$, such thing can be coded by a subset of $\alpha$. Then you guess all the names. Finally, $\kappa$-cc forcing doesn't change stationarity wrt $\kappa$. $\endgroup$
    – Otto
    Commented 19 hours ago
  • $\begingroup$ Yes, I'm interested in if some stronger forms of diamond remains (as well as what stronger diamonds cannot be introduced by such forcings). $\endgroup$
    – Yujun Wei
    Commented 5 hours ago
  • $\begingroup$ I have doubts, such a forcing can even destroy supercompactess of the cardinal ( even it's inaccessibility), just add $\kappa$ Cohen reals. You need first preserve enough supercompactess, then under mild conditions, each embedding is an extension of ground model embedding and then usually one gets what you want. $\endgroup$
    – Mohammad Golshani
    Commented 4 hours ago

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