4 of 4 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 $$M[G]$$?

Here

• $$M$$ 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.