The upper attic
From Cantor's 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.
- The Kunen inconsistency: Reinhardt cardinal, super Reinhardt cardinal, Berkeley cardinal
- Rank into rank axioms, I0 axiom and strengthenings
- The wholeness axioms
- n-fold variants of hugeness, extendibility, supercompactness, strongness, etc...
- huge cardinal, superhuge cardinal, ultrahuge cardinal, 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 cardinal, Woodin for supercompactness cardinal
- Vopěnka's principle
- extendible cardinal, $\alpha$-extendible cardinal
- 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
- 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
- super Ramsey cardinal
- strongly Ramsey cardinal
- Ramsey cardinal, Jónsson cardinal, Rowbottom cardinal, virtually Ramsey cardinal
- almost Ramsey cardinal
- $\omega_1$-Erdős cardinal and $\gamma$-Erdős cardinals for uncountable $\gamma$, Chang's conjecture
- $\omega_1$-iterable cardinal
- $0^\sharp$ (zero-sharp), existence of Silver indiscernibles
- Erdős cardinal, and the $\alpha$-Erdős hierarchy for countable $\alpha$
- the $\alpha$-iterable cardinals hierarchy for $1\leq\alpha<\omega_1$
- remarkable cardinal
- weakly Ramsey cardinal
- ineffable cardinal, weakly ineffable cardinal, and the $n$-ineffable cardinals hierarchy; completely ineffable cardinal
- subtle cardinal
- ethereal cardinal
- superstrongly unfoldable cardinal, strongly uplifting cardinal
- weakly superstrong cardinal
- unfoldable cardinal, strongly unfoldable cardinal
- indescribable hierarchy, totally indescribable cardinal
- weakly compact cardinal
- The positive set theory $\text{GPK}^+_\infty$
- Mahlo cardinal, $1$-Mahlo, the $\alpha$-Mahlo hierarchy, hyper-Mahlo cardinals
- uplifting cardinal, pseudo 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 and hyper-inaccessible cardinals
- Grothendieck universe axiom, equivalent to the existence of a proper class of inaccessible cardinals
- inaccessible cardinal, weakly 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