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Let $\mathrm{ZF^-}$ be $\mathrm{ZF}$ minus power set, and let $\beta_0$ be the ordinal for ramified analysis so that $\mathrm{L}_{\beta_0}$ is the least $\mathrm{L}$-model of $\mathrm{ZF}^-$. Clearly, set theory $\mathrm{ZF^-+V=L_{\beta_0}}$ only has countably many sets, which are all countable.

What is the least ordinal $\chi$ such that $\mathrm{L}_\chi$ contains a surjection from $\omega$ to $\mathrm{L}_{\beta_0}$?

For $\beta_0$, cfr. P. D. Welch The Ramified Analytical Hierarchy using Extended Logics, pp. 3-4.

Let $\mathrm{ZF^-}$ be $\mathrm{ZF}$ minus power set, and let $\beta_0$ be the ordinal for ramified analysis so that $\mathrm{L}_{\beta_0}$ is the least $\mathrm{L}$-model of $\mathrm{ZF}^-$. Clearly, set theory $\mathrm{ZF^-+V=L_{\beta_0}}$ only has countably many sets, which are all countable.

What is the least ordinal $\chi$ such that $\mathrm{L}_\chi$ has a surjection from $\omega$ to $\mathrm{L}_{\beta_0}$?

For $\beta_0$, cfr. P. D. Welch The Ramified Analytical Hierarchy using Extended Logics, pp. 3-4.

Let $\mathrm{ZF^-}$ be $\mathrm{ZF}$ minus power set, and let $\beta_0$ be the ordinal for ramified analysis so that $\mathrm{L}_{\beta_0}$ is the least $\mathrm{L}$-model of $\mathrm{ZF}^-$. Clearly, set theory $\mathrm{ZF^-+V=L_{\beta_0}}$ only has countably many sets, which are all countable.

What is the least ordinal $\chi$ such that $\mathrm{L}_\chi$ contains a surjection from $\omega$ to $\mathrm{L}_{\beta_0}$?

For $\beta_0$, cfr. P.D. Welch The Ramified Analytical Hierarchy using Extended Logics, pp. 3-4.

Let $\mathrm{ZF^-}$ be $\mathrm{ZF}$ minus power set, and let $\beta_0$ be the ordinal for ramified analysis so that $\mathrm{L}_{\beta_0}$ is the least $\mathrm{L}$-model of $\mathrm{ZF}^-$. Clearly, set theory $\mathrm{ZF^-+V=L_{\beta_0}}$ only has countably many sets, which are all countable.

What is the least ordinal $\chi$ such that $\mathrm{L}_\chi$ contains a surjection from $\omega$ to $\mathrm{L}_{\beta_0}$?

For $\beta_0$, cfr. P. D. Welch The Ramified Analytical Hierarchy using Extended Logics, pp. 3-4.

Let $\mathrm{ZF^-}$ be $\mathrm{ZF}$ minus power set, and let $\beta_0$ be the ordinal for ramified analysis so that $\mathrm{L}_{\beta_0}$ is the least $\mathrm{L}$-model of $\mathrm{ZF}^-$. Clearly, set theory $\mathrm{ZF^-+V=L_{\beta_0}}$ only has countably many sets, all of whom are countable.

What is the least ordinal $\chi$ such that $\mathrm{L}_\chi$ contains a surjection from $\omega$ to $\mathrm{L}_{\beta_0}$?

For $\beta_0$, cfr. P.D. Welch The Ramified Analytical Hierarchy using Extended Logics, pp. 3-4.

Let $\mathrm{ZF^-}$ be $\mathrm{ZF}$ minus power set, and let $\beta_0$ be the ordinal for ramified analysis so that $\mathrm{L}_{\beta_0}$ is the least $\mathrm{L}$-model of $\mathrm{ZF}^-$. Clearly, set theory $\mathrm{ZF^-+V=L_{\beta_0}}$ only has countably many sets, which are all countable.

What is the least ordinal $\chi$ such that $\mathrm{L}_\chi$ contains a surjection from $\omega$ to $\mathrm{L}_{\beta_0}$?

For $\beta_0$, cfr. P.D. Welch The Ramified Analytical Hierarchy using Extended Logics, pp. 3-4.