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retitled; added (forcing) and (lo.logic) tags; gave more explicit sources; defined some terms used in the question. (Possibly a lost cause, but in tune with some of the comments to the question.)

Preservation of $\diamondsuit$ by ccc forcings of size $\leq \omega_1$

This is essentially exercise H8 (p.248) of Kunen's Set Theory: An Introduction to Independence Proofs (old edition), or exercise IV.7.58 (p.307) of Kunen's Set Theory (new edition).

Suppose $P$ is a notion of forcing in $M$ such that $\left | P \right | \leq \omega_{1}$ and $P$ is ccc. Suppose further $\Diamond$ holds in $M$. How does one show that $\Diamond$ also holds $M[G]$?


  • $M$ is a countable transitive model (of $\mathsf{ZFC}$), and $M[G]$ is a generic extension of $M$ by the forcing $P$.

  • $P$ being ccc (countable chain condition) means that all antichains (sets of pairwise incompatible conditions) in $P$ are countable.

  • $\diamondsuit$ is the usual diamond principle:

    There is sequence $\langle A_\alpha : \alpha < \omega_1 \rangle$ such that $A_\alpha \subseteq \alpha$ for each $\alpha < \omega_1$, and for each $A \subseteq \omega_1$ the set $$\{ \alpha < \omega_1 : A \cap \alpha = A_\alpha \}$$ is stationary.