Diamond on ω1

A diamond sequence on ${\omega }_1$ is a sequence $(A_{\alpha }{)}_{\alpha <{\omega }_1}$ such that $A_{\alpha }\subseteq \alpha$ for every $\alpha$, and for every $A\subseteq {\omega }_1$ the set

$S_A=\{\alpha <{\omega }_1:A\cap \alpha =A_{\alpha }\}$

is stationary. Here ${\omega }_1$ is the first uncountable ordinal of The first uncountable ordinal ${\omega }_1:=\mathrm{\aleph }(\omega )$, and stationary means meeting every club, as in The club filter and nonstationary ideal and Closed unbounded subsets of ordinals. Thus the quantifiers are: for each $A\subseteq {\omega }_1$ and each club $C\subseteq {\omega }_1$, there is $\alpha \in C$ with $A\cap \alpha =A_{\alpha }$.

The principle $\mathrm{♢}$ asserts the existence of such a sequence; it is an additional hypothesis whenever used here. At zero the subset requirement forces $A_0=\mathrm{\varnothing }$; at one the two possible guesses are $\mathrm{\varnothing }$ and $\{0\}$. Stationarity is required also for the targets $A=\mathrm{\varnothing }$ and $A={\omega }_1$. Merely requiring an unbounded set of guesses is a different condition and is not the definition. No restriction to countable targets is intended.