= Solution
For every $\beta<\omega_1$, choose an <injection> $e_\beta:\beta\to\omega$. Define the <Ulam matrix on omega-one> by
$$
\boxed{A_{\alpha,n}=\{\beta<\omega_1:\alpha<\beta\text{ and }e_\beta(\alpha)=n\}.}
$$
For a fixed $\alpha$, every $\beta>\alpha$ belongs to exactly one of these sets, so their union is the entire tail $(\alpha,\omega_1)$. Its complement is the countable <ordinal> $\alpha+1$. For distinct $\alpha,\alpha'$ and fixed $n$, membership in both sets would give $e_\beta(\alpha)=e_\beta(\alpha')=n$ for some $\beta$ above both, contradicting injectivity. This verifies both requested properties.
Back to article page
