The upper attic
From Cantor's Attic
Revision as of 20:41, 8 October 2019 by Master (Talk | contribs) (Misrepresents the other of large cardinals. Hyperhuge=2-Supercompact>2-Strong)
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, $C^{(n)}$-I3, $C^{(n)}$-I1, 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 (=Woodin for supercompactness) cardinal
- Vopěnka's principle
- $C^{(n)}$-extendible hierarchy
- extendible cardinal, $\alpha$-extendible cardinal
- Woodin for strong compactness
- hypercompact cardinal
- supercompact cardinal, $\lambda$-supercompact cardinal
- strongly compact cardinal $\lambda$-strongly compact cardinal
- nearly supercompact and nearly strongly compact cardinals
- indestructible weakly compact cardinal
- The proper forcing axiom and Martin's maximum
- subcompact cardinal
- $C^{(n)}$-superstrong hierarchy
- superstrong cardinal
- Shelah cardinal
- The axiom of determinacy and its projective counterpart
- Woodin cardinal
- strongly tall cardinal
- strong cardinal and the $\theta$-strong and hypermeasurability hierarchies, tall cardinal, $\theta$-tall hierarchy
- Nontrivial Mitchell rank, $o(\kappa)=1$, $o(\kappa)=\kappa^{++}$
- $0^\dagger$ (zero-dagger)
- measurable cardinal, weakly 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
- Ramsey cardinal, Jónsson cardinal, Rowbottom cardinal, virtually 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 cardinal (=superstrongly unfoldable)
- weakly superstrong cardinal
- unfoldable cardinal, strongly unfoldable cardinal
- $η$-shrewd, shrewd, $\mathcal{A}$-$η$-shrewd, $\mathcal{A}$-shrewd cardinals
- indescribable hierarchy, totally indescribable cardinal
- $\Sigma_n$-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