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clique idea – Large cardinals and reflection properties Answer

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clique idea – Large cardinals and reflection properties

There is definitely a kind of cardinal that satisfies this, referred to as the “stationarily superhuge” cardinals. A cardinal $kappa$ is named stationarily superhuge if the ${lambda|kappatext{ is gigantic with goal }lambda}$ is stationary. If $kappa$ is stationarily superhuge, $V_kappaprec V$, and furthermore $L_kappaprec L$. The intuition for that is that $V_lambdaprec V$ is membership, assuming the actuality of a stationarily superhuge cardinal. Then $V_kappavDashphi$ if and provided that $MvDash(V_lambdavDashphi)$ if and provided that $V_lambdavDashphi$. Setting $lambda$ is rectify, we have now $V_lambda$ displays $phi$.

An analogous dispute works for $L_kappa$, and for $H_kappa$ (Or extra merely, as $kappa$ is superhuge and so inaccessible, $H_kappa=V_kappa$). Moreover, the clique of such cardinals figure a regular touchstone beneath $kappa$. Let $D=kappain j(X)$. Then $MvDash(j(kappa)textual content{ is reflecting})$, and so $Uin D$, the place $U={lambda<kappa|lambdatext{ is reflecting}}$.

Reference: https://www.jstor.org/steady/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)

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