Diamond principle

Remarks

Example: Jensen proved that $\diamondsuit$ holds in the constructible universe $L$, so it is consistent with ZFC if ZFC is consistent. Non-example: the sequence with $A_\alpha = \varnothing$ for all $\alpha$ is not a $\diamondsuit$-sequence. For $A = \omega_1$, the set $\set{\alpha : \alpha \cap \omega_1 = \varnothing} = \set{0}$ is not stationary. $\diamondsuit$ implies the continuum hypothesis: each $A \subseteq \omega$ equals $A \cap \alpha = A_\alpha$ for some $\alpha > \omega$, so there are at most $\aleph_1$ subsets of $\omega$. $\diamondsuit$ implies that there is a Suslin tree, and so it settles Suslin's problem negatively. More generally, $\diamondsuit_\kappa(S)$ is defined for a regular uncountable cardinal $\kappa$ and a stationary $S \subseteq \kappa$, with guesses only at points of $S$.