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There is actually a type of cardinal that satisfies this, called the "stationarily superhuge" cardinals. A cardinal $\kappa$ is called stationarily superhuge if the $\{\lambda|\kappa\text{ is huge with target }\lambda\}$ is stationary. If $\kappa$ is stationarily superhuge, $V_\kappa\prec V$, and moreover $L_\kappa\prec L$. The reason for this is that $\{\lambda|V_\lambda\prec V\}$ is club, assuming the existence of a stationarily superhuge cardinal. Then $V_\kappa\vDash\phi$ if and only if $M\vDash(V_\lambda\vDash\phi)$ if and only if $V_\lambda\vDash\phi$. Setting $\lambda$ is correct, we have $V_\lambda$ reflects $\phi$.

A similar argument works for $L_\kappa$, and for $H_\kappa$ (Or more simply, as $\kappa$ is superhuge and so inaccessible, $H_\kappa=V_\kappa$). Moreover, the set of such cardinals form a normal measure beneath $\kappa$. Let $D=\{X\subseteq\kappa|\kappa\in j(X)\}$. Then $M\vDash(j(\kappa)\text{ is reflecting})$, and so $U\in D$, where $U=\{\lambda<\kappa|\lambda\text{ is reflecting}\}$.

Reference: https://www.jstor.org/stable/2274094 (Julius B. Barbanel, Carlos A. Diprisco and It Beng Tan: Many-Times Huge and Superhuge Cardinals, The Journal of Symbolic Logic, Vol. 49, No. 1 (Mar., 1984), pp. 112-122. https://projecteuclid.org/euclid.jsl/1183741478 https://doi.org/10.2307/2274094)

There is actually a type of cardinal that satisfies this, called the "stationarily superhuge" cardinals. A cardinal $\kappa$ is called stationarily superhuge if the $\{\lambda|\kappa\text{ is huge with target }\lambda\}$ is stationary. If $\kappa$ is stationarily superhuge, $V_\kappa\prec V$, and moreover $L_\kappa\prec L$. The reason for this is that $\{\lambda|V_\lambda\prec V\}$ is club, assuming the existence of a stationarily superhuge cardinal. Then $V_\kappa\vDash\phi$ if and only if $M\vDash(V_\lambda\vDash\phi)$ if and only if $V_\lambda\vDash\phi$. Setting $\lambda$ is correct, we have $V_\lambda$ reflects $\phi$.

A similar argument works for $L_\kappa$, and for $H_\kappa$ (Or more simply, as $\kappa$ is superhuge and so inaccessible, $H_\kappa=V_\kappa$). Moreover, the set of such cardinals form a normal measure beneath $\kappa$. Let $D=\{X\subseteq\kappa|\kappa\in j(X)\}$. Then $M\vDash(j(\kappa)\text{ is reflecting})$, and so $U\in D$, where $U=\{\lambda<\kappa|\lambda\text{ is reflecting}\}$.

Reference:

• Julius B. Barbanel, Carlos A. Diprisco and It Beng Tan: Many-Times Huge and Superhuge Cardinals, The Journal of Symbolic Logic, Vol. 49, No. 1 (Mar., 1984), pp. 112-122. https://doi.org/10.2307/2274094 https://www.jstor.org/stable/2274094

There is actually a type of cardinal that satisfies this, called the "stationarily superhuge" cardinals. A cardinal $\kappa$ is called stationarily superhuge if the $\{\lambda|\kappa\,is\,huge\,with\,target\,\lambda\}$ is stationary. If $\kappa$ is stationarily superhuge, $V_\kappa\prec V$, and moreover $L_\kappa\prec L$. The reason for this is that $\{\lambda|V_\lambda\prec V\}$ is club, assuming the existence of a stationarily superhuge cardinal. Then $V_\kappa\vDash\phi$ if and only if $M\vDash(V_\lambda\vDash\phi)$ if and only if $V_\lambda\vDash\phi$. Setting $\lambda$ is correct, we have $V_\lambda$ reflects $\phi$.

A similar arguement works for $L_\kappa$, and for $H_\kappa$ (Or more simply, as $\kappa$ is superhuge and so inaccessible, $H_\kappa=V_\kappa$). Moreover, the set of such cardinals form a normal measure beneath $\kappa$. Let $D=\{X\subseteq\kappa|\kappa\in j(X)\}$. Then $M\vDash(j(\kappa)\,is \,reflecting)$, and so $U\in D$, where $U=\{\lambda<\kappa|\lambda\,is\,reflecting\}$.

Reference: https://www.jstor.org/stable/2274094?seq=1#metadata_info_tab_contents

There is actually a type of cardinal that satisfies this, called the "stationarily superhuge" cardinals. A cardinal $\kappa$ is called stationarily superhuge if the $\{\lambda|\kappa\text{ is huge with target }\lambda\}$ is stationary. If $\kappa$ is stationarily superhuge, $V_\kappa\prec V$, and moreover $L_\kappa\prec L$. The reason for this is that $\{\lambda|V_\lambda\prec V\}$ is club, assuming the existence of a stationarily superhuge cardinal. Then $V_\kappa\vDash\phi$ if and only if $M\vDash(V_\lambda\vDash\phi)$ if and only if $V_\lambda\vDash\phi$. Setting $\lambda$ is correct, we have $V_\lambda$ reflects $\phi$.

A similar argument works for $L_\kappa$, and for $H_\kappa$ (Or more simply, as $\kappa$ is superhuge and so inaccessible, $H_\kappa=V_\kappa$). Moreover, the set of such cardinals form a normal measure beneath $\kappa$. Let $D=\{X\subseteq\kappa|\kappa\in j(X)\}$. Then $M\vDash(j(\kappa)\text{ is reflecting})$, and so $U\in D$, where $U=\{\lambda<\kappa|\lambda\text{ is reflecting}\}$.

Reference: https://www.jstor.org/stable/2274094 (Julius B. Barbanel, Carlos A. Diprisco and It Beng Tan: Many-Times Huge and Superhuge Cardinals, The Journal of Symbolic Logic, Vol. 49, No. 1 (Mar., 1984), pp. 112-122. https://projecteuclid.org/euclid.jsl/1183741478 https://doi.org/10.2307/2274094)

There is actually a type of cardinal that satisfies this, called the "stationarily superhuge" cardinals. A cardinal $\kappa$ is called stationarily superhuge if the $\{\lambda|\kappa\,is\,huge\,with\,target\,\lambda\}$ is stationary. If $\kappa$ is stationarily superhuge, $V_\kappa\prec V$, and moreover $L_\kappa\prec L$. The reason for this is that $\{\lambda\,is\,inaccessible|V_\lambda\prec V\}$ is club, assuming the existence of a stationarily superhuge cardinal. Then $V_\kappa\vDash\phi$ if and only if $M\vDash(V_\lambda\vDash\phi)$ if and only if $V_\lambda\vDash\phi$. Setting $\lambda$ is reflecting, we have $V_\lambda$ reflects $\phi$.

A similar arguement works for $L_\kappa$, and for $H_\kappa$ (Or more simply, as $\kappa$ is superhuge and so inaccessible, $H_\kappa=V_\kappa$). Moreover, the set of such cardinals form a normal measure beneath $\kappa$. Let $D=\{X\subseteq\kappa|\kappa\in j(X)\}$. Then $M\vDash(j(\kappa)\,is \,reflecting)$, and so $U\in D$, where $U=\{\lambda<\kappa|\lambda\,is\,reflecting\}$. Moreover, $M\vDash (L_{j(\kappa)}\prec L)$, and so $S\in D$, where $S=\{\lambda<\kappa|L_\lambda\prec L\}$. As $D$ is an ultrafilter, $U\cap S\in D$, and so every stationarily superhuge cardinal bigger than many cardinals $\lambda$ such that $\lambda$ is both reflecting and $L_\lambda\prec L$.

Reference: https://www.jstor.org/stable/2274094?seq=1#metadata_info_tab_contents

There is actually a type of cardinal that satisfies this, called the "stationarily superhuge" cardinals. A cardinal $\kappa$ is called stationarily superhuge if the $\{\lambda|\kappa\,is\,huge\,with\,target\,\lambda\}$ is stationary. If $\kappa$ is stationarily superhuge, $V_\kappa\prec V$, and moreover $L_\kappa\prec L$. The reason for this is that $\{\lambda|V_\lambda\prec V\}$ is club, assuming the existence of a stationarily superhuge cardinal. Then $V_\kappa\vDash\phi$ if and only if $M\vDash(V_\lambda\vDash\phi)$ if and only if $V_\lambda\vDash\phi$. Setting $\lambda$ is correct, we have $V_\lambda$ reflects $\phi$.

A similar arguement works for $L_\kappa$, and for $H_\kappa$ (Or more simply, as $\kappa$ is superhuge and so inaccessible, $H_\kappa=V_\kappa$). Moreover, the set of such cardinals form a normal measure beneath $\kappa$. Let $D=\{X\subseteq\kappa|\kappa\in j(X)\}$. Then $M\vDash(j(\kappa)\,is \,reflecting)$, and so $U\in D$, where $U=\{\lambda<\kappa|\lambda\,is\,reflecting\}$.

Reference: https://www.jstor.org/stable/2274094?seq=1#metadata_info_tab_contents