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Denote $W=\lbrace R \mid R \hspace{0.1cm} \text{is a well-order on} \hspace{0.1cm} \omega \rbrace$ and $S=\lbrace \text{Ord}(R)\mid R \in W \rbrace$. The ZF-axioms guarantuee us that $W$ and $S$ are sets. Now let $\gamma:= \text{sup}(S)$. How does one prove that, without the use of the Axiom of Choice, that $\gamma$ is a cardinal such that $\omega < \gamma$?

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