# 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.

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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