# Difference between revisions of "Upper attic"

From Cantor's Attic

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* [[Mahlo]] cardinal | * [[Mahlo]] cardinal | ||

* [[ORD is Mahlo]] | * [[ORD is Mahlo]] | ||

− | * [[reflecting#inaccessible reflecting | inaccessible $\Sigma_2$-reflecting]], [[reflecting#inaccessible reflecting | inaccessible $\Sigma_n$-reflecting]] and [[reflecting#inaccessible reflecting | inaccessible reflecting]] cardinals | + | * [[reflecting#inaccessible reflecting | inaccessible $\Sigma_2$-reflecting]], [[reflecting#inaccessible reflecting | 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#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#Universes | Grothendieck universe axiom]], equivalent to the existence of a proper class of [[inaccessible]] cardinals |

## Revision as of 05:28, 30 December 2011

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
- extendible cardinal
- grand reflection cardinal
- supercompact cardinal
- strongly compact cardinal
- nearly supercompact and nearly strongly compact cardinals
- superstrong cardinal
- Woodin cardinal
- strong cardinal
- tall cardinal
- measurable cardinal
- weakly measurable cardinal
- strongly Ramsey cardinal
- weakly Ramsey cardinal
- iterably Ramsey cardinal
- Ramsey cardinal
- subtle cardinal
- $0^\sharp$
- Erdos cardinal, $\omega_1$-Erdos and the $\alpha$-Erdos hierarchy
- ineffible 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
- Transitive ZFC model
- the minimal model
- Con(ZFC) and the iterated consistency hierarchy

- down to the middle attic