If $W(\zeta)$ is an initial segment of an ordinal $\beta$ then can I be sure that $\zeta \in \beta$?
Is this definitely true?
2026-03-26 10:58:14.1774522694
Ordinals and initial segments
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Any proper initial segment of an ordinal is an element of that ordinal. (Obviously the whole ordinal itself is an initial segment which isn't an element of itself.)
Specifically, suppose $\alpha$ is an ordinal and $\Theta\subseteq\alpha$ is a proper initial segment of $\alpha$. Let $\theta$ be the least element of $\alpha$ not in $\Theta$. Then we have $\theta=\Theta$ (by the definition of ordinals) and $\theta\in\alpha$.