Berkeley

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A cardinal $\kappa$ is a Berkeley cardinal, if for any transitive set $M$ with $\kappa\in M$ and any ordinal $\alpha\lt\kappa$ there is an elementary embedding $j:M\prec M$ with $\alpha<\text{crit }j<\kappa$. These cardinals are defined in the context of ZF set theory without the axiom of choice.

The Berkeley cardinals were defined by W. Hugh Woodin in about 1992 at his set-theory seminar in Berkeley, with J. D. Hamkins, A. Lewis, D. Seabold, G. Hjorth and perhaps R. Solovay in the audience, among others, issued as a challenge to refute a seemingly over-strong large cardinal axiom. Nevertheless, the existence of these cardinals remains unrefuted in ZF.

If there is a Berkeley cardinal, then there is a forcing extension that forces that the least Berkeley cardinal has cofinality $\omega$. It seems that various strengthenings of the Berkeley property can be obtained by imposing conditions on the cofinality of $\kappa$ (The larger cofinality, the stronger theory is believed to be, up to regular $\kappa$).[1] If $\kappa$ is Berkeley and $a,\kappa\in M$ for $M$ transitive, then for any $\alpha\lt\kappa$, there is a $j: M\prec M$ with $\alpha\lt\text{crit }j\lt\kappa$ and $j(a)=a$.[1]

A cardinal $\kappa$ is called proto-Berkeley if for any transitive $M\ni\kappa$, there is some $j: M\prec M$ with $\text{crit }j\lt\kappa$. More generally, a cardinal is $\alpha$-proto-Berkeley if and only if for any transitive set $M\ni\kappa$, there is some $j: M\prec M$ with $\alpha\lt\text{crit }j\lt\kappa$, so that if $\delta\ge\kappa$, $\delta$ is also $\alpha$-proto-Berkeley. The least $\alpha$-proto-Berkeley cardinal is called $\delta_\alpha$.

We call $\kappa$ a club Berkeley cardinal if $\kappa$ is regular and for all clubs $C\subseteq\kappa$ and all transitive sets $M$ with $\kappa\in M$ there is $j\in \mathcal{E}(M)$ with $\mathrm{crit}(j) ∈ C$.[1]

We call $\kappa$ a limit club Berkeley cardinal if it is a club Berkeley cardinal and a limit of Berkeley cardinals.[1]

Relations

  • If $\kappa$ is the least Berkeley cardinal, then there is $\gamma\lt\kappa$ such that $(V_\gamma , V_{\gamma+1})\vDash\mathrm{ZF}_2 + \text{“There is a Reinhardt cardinal witnessed by $j$ and an $\omega$-huge above $\kappa_\omega(j)”$}$.[1]
  • For every $\alpha$, $\delta_\alpha$ is Berkeley. Therefore $\delta_\alpha$ is the least Berkeley cardinal above $\alpha$.[1]
  • In particular, the least proto-Berkeley cardinal $\delta_0$ is also the least Berkeley cardinal.[1]
  • If $\kappa$ is a limit of Berkeley cardinals, then $\kappa$ is not among the $\delta_\alpha$.[1]
  • Each club Berkeley cardinal is totally Reinhardt.[1].
  • The relation between Berkeley cardinals and club Berkeley cardinals is unknown.[1]
  • If $\kappa$ is a limit club Berkeley cardinal, then $(V_\kappa , V_{\kappa+1})\vDash\text{“There is a Berkeley cardinal that is super Reinhardt”}$.[1] Moreover, the class of such cardinals are stationary.

The structure of $L(V_{\delta+1})$

If $\delta$ is a singular Berkeley cardinal, $DC(cf(\delta)^+)$, and $\delta$ is a limit of cardinals themselves limits of extendible cardinals, then the structure of $L(V_{\delta+1})$ is similar to the structure of $L(V_{\lambda+1})$ under the assumption $\lambda$ is $I0$; i.e. there is some $j: L(V_{\lambda+1})\prec L(V_{\lambda+1})$. For example, $\Theta=\Theta_{V_{\delta+1}}^{L(V_{\delta+1})}$, then $\Theta$ is a strong limit in $L(V_{\delta+1})$, $\delta^+$ is regular and measurable in $L(V_{\delta+1})$, and $\Theta$ is a limit of measurable cardinals.

References

  1. Bagaria, Joan. Large Cardinals beyond Choice. , 2017. www   bibtex
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