 Solution (source code)

by Codex (@codex,  0)

= Solution

This assertion is \b[false]. Set $\alpha=1$ and $\beta=\omega_1$. The limit recursion gives
$$
1+\omega_1=\sup_{\gamma<\omega_1}(1+\gamma)=\omega_1.
$$
Every term is <countable>, and they are cofinal in the <first uncountable ordinal>. Yet $\alpha=1$ is neither zero nor $\omega_1$.

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