# Difference between revisions of "Upper attic"

From Cantor's Attic

Line 14: | Line 14: | ||

* [[huge]] cardinal | * [[huge]] cardinal | ||

* [[huge#Almost huge | almost huge]] cardinal | * [[huge#Almost huge | almost huge]] cardinal | ||

+ | * [[Vopenka | Vopěnka]] cardinal, [[Vopenka#Vopěnka's principle | Vopěnka's principle] | ||

* [[extendible]] cardinal | * [[extendible]] cardinal | ||

* [[grand reflection]] cardinal | * [[grand reflection]] cardinal | ||

Line 19: | Line 20: | ||

* [[strongly compact]] cardinal | * [[strongly compact]] cardinal | ||

* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals | * [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals | ||

+ | * [[subcompact]] cardinal | ||

* [[strong#Superstrong | superstrong]] cardinal | * [[strong#Superstrong | superstrong]] cardinal | ||

+ | * [[Shelah]] cardinal | ||

* [[Woodin]] cardinal | * [[Woodin]] cardinal | ||

* [[strong]] cardinal | * [[strong]] cardinal | ||

Line 29: | Line 32: | ||

* [[Ramsey#Weakly_Ramsey | weakly Ramsey]] cardinal | * [[Ramsey#Weakly_Ramsey | weakly Ramsey]] cardinal | ||

* [[Ramsey#Iterably_Ramsey | iterably Ramsey]] cardinal | * [[Ramsey#Iterably_Ramsey | iterably Ramsey]] cardinal | ||

− | * [[Ramsey]] cardinal | + | * [[Ramsey]] cardinal |

− | * [[ | + | * [[Rowbottom]] cardinal |

+ | * [[Jónsson]] cardinal | ||

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

* [[Erdos]] cardinal, [[Erdos | $\omega_1$-Erdos]] and the [[Erdos | $\alpha$-Erdos]] hierarchy | * [[Erdos]] cardinal, [[Erdos | $\omega_1$-Erdos]] and the [[Erdos | $\alpha$-Erdos]] hierarchy | ||

+ | * [[remarkable]] cardinals | ||

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

+ | * [[subtle]] cardinal | ||

+ | * [[ethereal]] cardinal | ||

* [[unfoldable]] cardinal, [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal | * [[unfoldable]] cardinal, [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal | ||

* [[indescribable#Totally indescribable | totally indescribable]] cardinal | * [[indescribable#Totally indescribable | totally indescribable]] cardinal | ||

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* [[Transitive ZFC model]] | * [[Transitive ZFC model]] | ||

* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]] | * the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]] | ||

− | * [[Con ZFC | Con(ZFC)]] and | + | * [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]] |

* down to [[the middle attic]] | * down to [[the middle attic]] |

## Revision as of 13:59, 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, [[Vopenka#Vopěnka's principle | 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
- $0^\sharp$
- Erdos cardinal, $\omega_1$-Erdos and the $\alpha$-Erdos hierarchy
- remarkable cardinals
- ineffible cardinal
- subtle cardinal
- ethereal cardinal
- unfoldable cardinal, strongly unfoldable cardinal
- totally indescribable cardinal
- indescribable cardinal
- weakly compact cardinal
- $1$-Mahlo, the $\alpha$-Mahlo hierarchy and hyper-Mahlo cardinals
- Mahlo cardinal
- ORD is Mahlo
- inaccessible $\Sigma_2$-reflecting, inaccessible $\Sigma_n$-reflecting and inaccessible reflecting cardinals
- $1$-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, also known as strongly inaccessible
- weakly inaccessible cardinal
- worldly cardinal and the $\alpha$-wordly hierarchy, hyper-worldly cardinal
- the transitive model universe axiom
- Transitive ZFC model
- the minimal transitive model
- Con(ZFC) and $\text{Con}^\alpha(\text{ZFC})$, the iterated consistency hierarchy

- down to the middle attic