# Difference between revisions of "Upper attic"

From Cantor's Attic

Line 35: | Line 35: | ||

* [[Rowbottom]] cardinal | * [[Rowbottom]] cardinal | ||

* [[Jonsson | Jónsson]] cardinal | * [[Jonsson | Jónsson]] cardinal | ||

+ | * [[Erdos | $\omega_1$-Erdős]] and [[Erdos | $\gamma$$-Erdős]] for uncountable $\gamma$ | ||

* [[zero sharp | $0^\sharp$]] | * [[zero sharp | $0^\sharp$]] | ||

− | * [[Erdos | Erdős]] cardinal, [[Erdos | $\ | + | * [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$ |

* [[remarkable]] cardinal | * [[remarkable]] cardinal | ||

* [[ineffible]] cardinal | * [[ineffible]] cardinal |

## Revision as of 14:23, 3 January 2012

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
- $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$
- rank+1 into rank+1 cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$
- rank into rank cardinal $j:V_\lambda\to V_\lambda$
- The wholeness axiom
- super $n$-huge cardinal
- superhuge cardinal
- huge cardinal
- almost huge cardinal
- Vopěnka cardinal, Vopěnka's principle
- extendible cardinal
- grand reflection cardinal
- supercompact cardinal
- strongly compact cardinal
- nearly supercompact and nearly strongly compact cardinals
- subcompact cardinal
- superstrong cardinal
- Shelah cardinal
- Woodin cardinal
- strong cardinal
- tall cardinal
- $0^\dagger$
- measurable cardinal
- weakly measurable cardinal
- strongly Ramsey cardinal
- weakly Ramsey cardinal
- iterably Ramsey cardinal
- Ramsey cardinal
- Rowbottom cardinal
- Jónsson cardinal
- $\omega_1$-Erdős and $\gamma$$-Erdős]] for uncountable $\gamma$ * [[zero sharp | $0^\sharp$]] * [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$ * [[remarkable]] cardinal * [[ineffible]] cardinal * [[subtle]] cardinal * [[ethereal]] cardinal * [[unfoldable]] cardinal, [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal * [[indescribable#Totally indescribable | totally indescribable]] cardinal * [[indescribable]] cardinal * [[weakly compact]] cardinal * [[Mahlo#Hyper-Mahlo | $1$-Mahlo]], the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy and [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals * [[Mahlo]] cardinal * [[ORD is Mahlo]] * [[reflecting#Inaccessible reflecting cardinal | inaccessible $\Sigma_2$-reflecting]], [[reflecting#Inaccessible reflecting cardinal | inaccessible $\Sigma_n$-reflecting]] and [[reflecting#Inaccessible reflecting cardinal| inaccessible reflecting]] cardinals * [[inaccessible#Degrees of inaccessibility | $1$-inaccessible]], the [[inaccessible#Degrees of inaccessibility | $\alpha$-inaccessible]] hierarchy and [[inaccessible#Hyper-inaccessible | hyper-inaccessible]] cardinals * [[inaccessible#Universes | Grothendieck universe axiom]], equivalent to the existence of a proper class of [[inaccessible]] cardinals * [[inaccessible]] cardinal, also known as strongly inaccessible * [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal * [[worldly]] cardinal and the [[worldly#Degrees of worldliness | $\alpha$-wordly]] hierarchy, [[worldly#Degrees of worldliness | hyper-worldly]] cardinal * the [[Transitive ZFC model#Transitive model universe axiom | transitive model universe axiom]] * [[Transitive ZFC model]] * the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]] * [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$, the iterated consistency hierarchy

- down to the middle attic