Berkeley cardinals are defined as \(\kappa \) such that 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 \(\alpha \lt \text{crit} \space j \lt \kappa\). These cardinals are defined in the context of ZF set theory without the axiom of choice.[1]
A cardinal \(\kappa \) is proto-Berkeley if for any transitive set \(M \ni \kappa\) there is some \(j: M \prec M\) with \(\text{crit} j \lt \kappa \), More generally, a cardinal is \(\alpha \)-proto-Berkeley iff for any transitive set \(M \ni \kappa\) there is some \(j: M \prec M\) with \(\alpha \) < crit j < \(\kappa \), so that if \(\delta \geq \kappa\), \(\delta\) is also \(\alpha\)-proto Berkeley. The least proto Berkeley cardinal is called \(\delta_{\alpha}\).
A cardinal \(\kappa\) is 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 \(\text{crit} j \in C \). A club Berkeley cardinal can be a limit club Berkeley if it is both a club Berkeley and a limit of Berkeleys.
We call \(\kappa\) a limit club Berkeley cardinal if it is a club Berkeley cardinal and a limit of Berkeley cardinals
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Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation · Absolute infinity
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function) · \(\omega_1^\mathfrak{Ch}\) (Omega one of chess) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\gamma,\zeta,\Sigma\) (Infinite time Turing machine ordinals) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Hardy hierarchy · Slow-growing hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Weak Buchholz's function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Weak Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function · Arai's psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · Berkeley cardinal · (Club Berkeley cardinal · limit club Berkeley cardinal)
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)