The upper attic

Welcome to the upper attic, the transfinite realm of large cardinals, the higher infinite, carrying us upward from the merely inaccessible and indescribable to the subtle and endlessly extendible concepts beyond, towards the calamity of inconsistency.

• Berkeley cardinal, club Berkeley, limit club Berkeley cardinal
• weakly Reinhardt, Reinhardt, super Reinhardt, totally Reinhardt cardinal
• the Kunen inconsistency
• rank into rank axioms ($I3$=$E_0$, $IE^\omega$, $IE$, $I2$=$E_1$, $E_i$, $I1$=$E_ω$ plus $m$-$C^{(n)}$-$E_i$), $\omega$-fold variants, I0 axiom and strengthenings
• The wholeness axioms, axioms $\mathrm{I}_4^n$
• $n$-fold variants of hugeness (plus $C^{(n)}$ variants), extendibility, supercompactness, strongness, etc…
• almost huge, huge, huge*, super almost huge, superhuge, ultrahuge, 2-superstrong cardinal
• high-jump cardinal, almost high-jump cardinal, super high-jump cardinal, high-jump with unbounded excess closure cardinal
• Shelah for supercompactness
• Vopěnka scheme, Vopěnka principle, Vopěnka-scheme cardinal, Vopěnka (=Woodin for supercompactness) cardinal
• $\alpha$-extendible hierarchy, extendible cardinal, $C^{(n)}$-extendible hierarchy
• Woodin for strong compactness
• enhanced $\lambda$-supercompact cardinals, enhanced supercompact cardinal, $\lambda$-hypercompact cardinals, hypercompact cardinal
• $\lambda$-supercompact cardinals, supercompact cardinal, $C^{(n)}$-supercompact cardinals
• $\lambda$-strongly compact cardinals, strongly compact cardinal
• nearly supercompact and nearly strongly compact cardinals
• indestructible weakly compact cardinal
• The proper forcing axiom and Martin’s maximum
• subcompact cardinal
• superstrong cardinal, $C^{(n)}$-superstrong hierarchy
• Shelah cardinal
• The axiom of determinacy and its projective counterpart
• Woodin cardinal
• strongly tall cardinal
• the $\theta$-strong, hypermeasurability and $\theta$-tall hierarchies, tall and strong cardinals
• Nontrivial Mitchell rank, $o(\kappa)=1$, $o(\kappa)=\kappa^{++}$
• $0^\dagger$ (zero-dagger)
• weakly measurable cardinal, measurable cardinal
  • singular Jónsson cardinal
  • $κ^+$-filter property, strategic $(\omega+1)$-Ramsey cardinal, strategic fully Ramsey cardinal, $ω_1$-very Ramsey cardinal, $κ$-very Ramsey cardinal
• $κ$-filter property, fully Ramsey (=$κ$-Ramsey) cardinal
• strongly Ramsey cardinal, strongly Ramsey M-rank, super Ramsey cardinal, super Ramsey M-rank
• $\alpha$-filter property, $\alpha$-Ramsey cardinal (for $\omega < \alpha < \kappa$), almost fully Ramsey (=$<κ$-Ramsey) cardinal
• $\Pi_\alpha$-Romsey, completely Romsey (=$ω$-very Ramsey), $\alpha$-hyper completely Romsey, super completely Romsey cardinals
• $\alpha$-Mahlo–Ramsey hierarchy
• Ramsey M-rank
• virtually Ramsey cardinal, Jónsson cardinal, Rowbottom cardinal, Ramsey cardinal
• $\alpha$-weakly Erdős cardinals, greatly Erdős cardinal
• almost Ramsey cardinal
• $\omega_1$-Erdős cardinal and $\gamma$-Erdős cardinals for uncountable $\gamma$, Chang’s conjecture
• $\omega_1$-iterable cardinal, $(\omega, \omega_1)$-Ramsey cardinal
• $0^\sharp$ (zero-sharp), existence of Silver indiscernibles
• Silver cardinal
• the $\alpha$-Erdős, $\alpha$-iterable and $(\omega, \alpha)$-Ramsey hierarchy for countable infinite $\alpha$
• $\omega$-Erdős cardinal, weakly remarkable cardinal that is not remarkable
• virtually rank-into-rank cardinal
• the $n$-iterable and virtually $n$-huge* hierarchy
• virtually Shelah for supercompactness cardinal
• virtually extendible (=$2$-remarkable), virtually $C^{(n)}$-extendible (=$n+1$-remarkable) cardinals, completely remarkable cardinal, Generic Vopěnka’s Principle
• ($1$-)remarkable (=virtually supercompact), virtually measurable, strategic $\omega$-Ramsey cardinals, weak Proper Forcing Axiom
• weakly Ramsey (=$1$-iterable) cardinal, super weakly Ramsey cardinals, $\omega$-Ramsey cardinal
• completely ineffable cardinal (= $\omega$-filter property)
• the $n$-subtle, $n$-almost ineffable, $n$-ineffable cardinals’ hierarchy
• $n$-Ramsey, genuine $n$-Ramsey, normal $n$-Ramsey, $<\omega$-Ramsey cardinals
• weakly ineffable (=almost ineffable=genuine $0$-Ramsey) cardinal, ineffable (=normal $0$-Ramsey) cardinal
• subtle cardinal
• ethereal cardinal
• strongly uplifting (=superstrongly unfoldable) cardinal
• weakly superstrong cardinal
• unfoldable cardinal, strongly unfoldable cardinal
• $η$-shrewd, shrewd, $\mathcal{A}$-$η$-shrewd, $\mathcal{A}$-shrewd cardinals
• $\Sigma^m_n$- and $\Pi^m_n$-indescribable, totally indescribable, $η$-indescribable cardinals
• $\Sigma_n$-weakly compact cardinals, $\Sigma_\omega$-weakly compact cardinal, weakly compact (=$\Pi_1^1$-indescribable=$0$-Ramsey) cardinal
• The positive set theory $\text{GPK}^+_\infty$
• $\Sigma_n$-Mahlo cardinals, $\Sigma_\omega$-Mahlo cardinal, weakly Mahlo cardinal, (strongly) Mahlo cardinal, $1$-Mahlo, the $\alpha$-Mahlo hierarchy, hyper-Mahlo cardinals, $Ω^α$-Mahlo cardinals
• pseudo uplifting cardinal, uplifting cardinal
• $\text{Ord}$ is Mahlo
• $\Sigma_2$-reflecting, $\Sigma_n$-reflecting and reflecting cardinals
• Jäger’s collapsing functions and ρ-inaccessible ordinals
• $1$-inaccessible, the $\alpha$-inaccessible hierarchy, hyper-inaccessible cardinals, $Ω^α$-inaccessible cardinals
• Grothendieck universe axiom (the existence of a proper class of inaccessible cardinals)
• weakly inaccessible cardinal, (strongly) inaccessible cardinal
• Morse-Kelley set theory
• worldly cardinal and the $\alpha$-wordly hierarchy, hyper-worldly cardinal
• the transitive model universe axiom
• transitive model of $\text{ZFC}$
• the minimal transitive model
• $\text{Con(ZFC)}$ and $\text{Con}^\alpha(\text{ZFC})$, the iterated consistency hierarchy

• Zermelo-Fraenkel set theory

• down to the middle attic