View source for Upper attic

{{DISPLAYTITLE:The upper attic}}
[[File:CapePogueLighthouse_medium.jpg|thumb|Cape Pogue Lighthouse photo by Timothy Valentine]]
[[Category:Large cardinal axioms]]

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, [[Berkeley|club Berkeley]], [[Berkeley|limit club Berkeley]] cardinal
* [[Reinhardt|weakly Reinhardt]], [[Reinhardt]], [[Reinhardt|super Reinhardt]], [[Reinhardt|totally Reinhardt]] cardinal
* the '''[[Kunen inconsistency]]'''
* '''[[rank into rank]]''' axioms &#40;$I3$=$E_0$, $IE^\omega$, $IE$, $I2$=$E_1$, $E_i$, $I1$=$E_ω$ plus $m$-$C^{&#40;n)}$-$E_i$), [[N-fold_variants#.24.5Comega.24-fold_variants|$\omega$-fold variants]], the '''[[L of V_lambda+1|I0 axiom]]''' and its strengthenings
* The [[wholeness axioms]], [[I4|axioms $\mathrm{I}_4^n$]]
* [[n-fold variants|$n$-fold variants]] of hugeness &#40;plus $C^{&#40;n)}$ variants), extendibility, supercompactness, strongness, etc...
* [[huge|almost huge]], '''[[huge]]''', [[huge|huge*]], [[huge|super almost huge]], [[huge|superhuge]], [[huge|ultrahuge]], [[superstrong|2-superstrong]] cardinal
* [[high-jump]] cardinal, [[high-jump|almost high-jump]] cardinal, [[high-jump|super high-jump]] cardinal, [[high-jump|high-jump with unbounded excess closure]] cardinal
* [[Woodin#Shelah cardinals|Shelah for supercompactness]]
* [[Vopenka|Vopěnka scheme]], '''[[Vopenka|Vopěnka principle]]''', [[Vopenka#Vopěnka cardinals|Vopěnka-scheme]] cardinal, [[Vopenka#Vopěnka cardinals|Vopěnka]] &#40;=[[Woodin#Shelah cardinals|Woodin for supercompactness]]) cardinal
* [[extendible|$\alpha$-extendible]], '''[[extendible]]''', [[extendible|$C^{&#40;n)}$-extendible]], [[extendible|$A$-extendible]] cardinals
* [[Woodin|Woodin for strong compactness]]
<!--* [[grand reflection]] cardinal-->
* [[Supercompact#Enhanced supercompact cardinals|enhanced $\lambda$-supercompact]] cardinals, [[Supercompact#Enhanced supercompact cardinals|enhanced supercompact]] cardinal, [[hypercompact|$\lambda$-hypercompact]] cardinals, [[hypercompact]] cardinal
* [[supercompact|$\lambda$-supercompact]] cardinals, '''[[supercompact]]''' cardinal, [[supercompact|$C^{&#40;n)}$-supercompact]] cardinals
* [[strongly compact|$\lambda$-strongly compact]] cardinals, '''[[strongly compact]]''' cardinal
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact|nearly strongly compact]] cardinals
* [[Weakly_compact#Indestructibility of a weakly compact cardinal|indestructible weakly compact]] cardinal
* The '''[[proper forcing axiom]]''' and [[forcing#Proper forcing|Martin's maximum]]
* [[subcompact]] cardinal
* [[superstrong]] cardinal, [[superstrong|$C^{&#40;n)}$-superstrong]] hierarchy
* [[Woodin|weakly hyper-Woodin]] cardinal, [[Shelah]] cardinal, [[Woodin|hyper-Woodin]] cardinal
* The '''[[axiom of determinacy]]''' and [[axiom of projective determinacy|its projective counterpart]]
* '''[[Woodin]]''' cardinal
* [[strongly tall]] cardinal
* the [[strong|$\theta$-strong]], [[strong#Hypermeasurable|hypermeasurability]], [[tall|$\theta$-tall]], [[strong|$\theta$-$A$-strong]], [[tall]], '''[[strong]]''', [[strong|$A$-strong]] cardinals
* Nontrivial [[Mitchell rank]], [[Mitchell rank|$o&#40;\kappa)=1$]], [[Mitchell rank|$o&#40;\kappa)=\kappa^{++}$]] 
*[[zero dagger| $0^\dagger$]] &#40;''zero-dagger'')
* [[weakly measurable]] cardinal, '''[[measurable]]''' cardinal
** singular [[Jonsson|Jónsson]] cardinal
** $κ^+$-[[filter property]], [[Ramsey|strategic $&#40;\omega+1)$-Ramsey]] cardinal, [[Ramsey|strategic fully Ramsey]] cardinal, [[Ramsey|$ω_1$-very Ramsey]] cardinal, [[Ramsey|$κ$-very Ramsey]] cardinal
* $κ$-[[filter property]], [[Ramsey|fully Ramsey]] &#40;=[[Ramsey|$κ$-Ramsey]]) cardinal 
* [[Ramsey#Strongly Ramsey cardinal|strongly Ramsey]] cardinal, [[Ramsey|strongly Ramsey M-rank]], [[Ramsey#Super Ramsey cardinal|super Ramsey]] cardinal, [[Ramsey|super Ramsey M-rank]]
* $\alpha$-[[filter property]], [[Ramsey|$\alpha$-Ramsey]] cardinal &#40;for $\omega < \alpha < \kappa$), [[Ramsey|almost fully Ramsey]] &#40;=[[Ramsey|$<κ$-Ramsey]]) cardinal 
* [[Ramsey|$\Pi_\alpha$-Ramsey]], [[Ramsey|completely Romsey]] &#40;=[[Ramsey|$ω$-very Ramsey]]), [[Ramsey|$\alpha$-hyper completely Romsey]], [[Ramsey|super completely Romsey]] cardinals
* [[Ramsey|$\alpha$-Mahlo–Ramsey]] hierarchy
* [[Ramsey|Ramsey M-rank]]
* [[Ramsey#Virtually Ramsey cardinal|virtually Ramsey]] cardinal, [[Jonsson|Jónsson]] cardinal, [[Rowbottom]] cardinal, '''[[Ramsey]]''' cardinal
* [[Erdos|$\alpha$-weakly Erdős]] cardinals, [[Erdos|greatly Erdős]] cardinal
* [[Ramsey#Almost Ramsey cardinal|almost Ramsey]] cardinal
* [[Erdos|$\omega_1$-Erdős]] cardinal and [[Erdos|$\gamma$-Erdős]] cardinals for uncountable $\gamma$, [[Chang's conjecture]]
* [[Ramsey#.24.5Calpha.24-iterable cardinal|$\omega_1$-iterable]] cardinal, [[Ramsey|$&#40;\omega, \omega_1)$-Ramsey]] cardinal
* '''[[zero sharp|$0^\sharp$]] &#40;''zero-sharp'')''', existence of [[Constructible universe#Silver indiscernibles|Silver indiscernibles]]
* [[Silver cardinal]]
* the [[Erdos|$\alpha$-'''Erdős''']], [[Ramsey#.24.5Calpha.24-iterable cardinal|$\alpha$-iterable]] and [[Ramsey|$&#40;\omega, \alpha)$-Ramsey]] hierarchy for countable infinite $\alpha$
* [[Erdos|$\omega$-Erdős]] cardinal, [[remarkable|weakly remarkable]] cardinal that is not remarkable
* [[rank into rank|virtually rank-into-rank]] cardinal
* the [[Ramsey#.24.5Calpha.24-iterable cardinal|$n$-iterable]] and [[huge|virtually $n$-huge*]] hierarchy
* [[Woodin|virtually Shelah for supercompactness]] cardinal
* [[extendible|virtually extendible]] &#40;=[[remarkable|$2$-remarkable]]), [[extendible|virtually $C^{&#40;n)}$-extendible]] &#40;=[[remarkable|$n+1$-remarkable]]) cardinals, [[remarkable|completely remarkable]] cardinal, [[Vopenka|Generic Vopěnka's Principle]]
* [[remarkable|&#40;$1$-)'''remarkable''']] &#40;=virtually supercompact), [[measurable|virtually measurable]], [[Ramsey|strategic $\omega$-Ramsey]] cardinals, [[proper forcing axiom|weak Proper Forcing Axiom]]
* [[Ramsey#.24.5Calpha.24-iterable cardinal|weakly Ramsey]] &#40;=$1$-iterable) cardinal, [[Ramsey|super weakly Ramsey]] cardinals, [[Ramsey|$\omega$-Ramsey]] cardinal
* [[completely ineffable]] cardinal &#40;= $\omega$-[[filter property]])
* [[Basic Theory of Elementary Embeddings]] &#40;[[BTEE|$\mathrm{BTEE}$]])
* [[ineffable#Helix|the $n$-subtle, $n$-almost ineffable, $n$-ineffable cardinals' hierarchy]]
* [[Ramsey|$n$-Ramsey]], [[Ramsey|genuine $n$-Ramsey]], [[Ramsey|normal $n$-Ramsey]], [[Ramsey|$<\omega$-Ramsey]] cardinals
* [[weakly ineffable]] &#40;=almost ineffable=genuine $0$-[[Ramsey]]) cardinal, [[ineffable]] &#40;=normal $0$-[[Ramsey]]) cardinal
* [[subtle]] cardinal
* [[ineffable#Ethereal cardinal|ethereal]] cardinal
* [[uplifting#Strongly Uplifting|strongly uplifting]] &#40;=[[unfoldable#Superstrongly Unfoldable|superstrongly unfoldable]]) cardinal
* [[weakly superstrong]] cardinal
* [[shrewd|$η$-shrewd]], [[unfoldable]], [[shrewd]] &#40;=[[unfoldable#Strongly Unfoldable|strongly unfoldable]] cardinal), [[shrewd|$\mathcal{A}$-$η$-$\mathcal{F}$-shrewd]], [[shrewd|$\mathcal{A}$-$η$-shrewd]], [[shrewd|$\mathcal{A}$-shrewd]] cardinals
* $\Sigma^m_n$- and '''$\Pi^m_n$-[[indescribable]]''', [[totally indescribable]], [[indescribable|$η$-indescribable]] cardinals
* uncountable cardinal with the [[tree property]], '''[[weakly compact]]''' &#40;=$\Pi_1^1$-[[indescribable]]=$0$-[[Ramsey]]) cardinal
* The [[Positive set theory|positive set theory]] $\text{GPK}^+_\infty$
* [[Mahlo#Hyper-Mahlo|$1$-Mahlo]], the [[Mahlo#Hyper-Mahlo|$\alpha$-Mahlo]] hierarchy, [[Mahlo#Hyper-Mahlo|hyper-Mahlo]] cardinals, [[Mahlo|$Ω^α$-Mahlo]] cardinals
* [[Mahlo|weakly Mahlo]] cardinal, &#40;strongly) '''[[Mahlo]]''' cardinal
* [[uplifting#pseudo uplifting cardinal|pseudo uplifting]] cardinal, [[uplifting]] cardinal
* [[ORD is Mahlo|$\text{Ord}$ is Mahlo]]<!-- apparently equiconsistent with a reflecting cardinal -->
* [[reflecting#Sigma_2 correct cardinals|$\Sigma_2$-reflecting]], [[reflecting|$\Sigma_n$-reflecting]] and [[reflecting]] cardinals
* [[Mahlo|$\Sigma_n$-Mahlo]], [[weakly compact|$\Sigma_n$-weakly compact]], [[Mahlo|$\Sigma_\omega$-Mahlo]] and [[weakly compact|$\Sigma_\omega$-weakly compact]]<!-- Really? In particular, are $\Sigma_\omega$ variants not stronger then ORD is Mahlo? Maybe $\Sigma_\omega$-weakly compact is even stronger than Mahlo? --> cardinals
* [[Jäger's collapsing functions and ρ-inaccessible ordinals]] 
* [[inaccessible#Degrees of inaccessibility|$1$-inaccessible]], the [[inaccessible#Degrees of inaccessibility|$\alpha$-inaccessible]] hierarchy,  [[inaccessible#Hyper-inaccessible|hyper-inaccessible]] cardinals, [[inaccessible|$Ω^α$-inaccessible]] cardinals
* [[inaccessible#Universes|Grothendieck universe axiom]] &#40;the existence of a proper class of [[inaccessible]] cardinals)
* [[inaccessible#Weakly inaccessible cardinal|weakly inaccessible]] cardinal, &#40;strongly) '''[[inaccessible]]''' cardinal
* [[Morse-Kelley set theory|Morse-Kelley]] set theory
* '''[[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|transitive model of $\text{ZFC}$]]
* the [[Transitive ZFC model#Minimal transitive model of ZFC|minimal transitive model]]
* '''[[Con ZFC|$\text{Con&#40;ZFC)}$]]''' and [[Con ZFC#Consistency hierarchy|$\text{Con}^\alpha&#40;\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy|iterated consistency hierarchy]]
* '''[[ZFC|Zermelo-Fraenkel]]''' set theory

* down to [[the middle attic]]