Difference between revisions of "Upper attic"

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{{DISPLAYTITLE:The upper attic}}
 
{{DISPLAYTITLE:The upper attic}}
 
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]
 
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]
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[[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.  
 
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, [[Kunen_inconsistency#Super_Reinhardt_cardinal | super Reinhardt]] cardinal, [[Berkeley]] cardinal
 
* The '''[[Kunen inconsistency]]''': [[Reinhardt]] cardinal, [[Kunen_inconsistency#Super_Reinhardt_cardinal | super Reinhardt]] cardinal, [[Berkeley]] cardinal
* '''[[Rank into rank]]''' cardinals $j:V_\lambda\to V_\lambda$, [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$, I0 cardinal [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]
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* '''[[Rank into rank]]''' axioms, [[L of V_lambda+1|I0 axiom]] and strengthenings
 
* The [[wholeness axioms]]
 
* The [[wholeness axioms]]
* '''[[huge|n-huge]]''' cardinal, [[huge | almost n-huge]] cardinal, [[huge|super-n-huge]] cardinal, [[superstrong|(n+1)-superstrong cardinal]]
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* [[n-fold variants]] of hugeness, extendibility, supercompactness, strongness, etc...
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* '''[[huge]]''' cardinal, [[huge|superhuge]] cardinal, [[huge|ultrahuge]] cardinal, [[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
 
* [[high-jump]] cardinal, [[high-jump|almost high-jump]] cardinal, [[high-jump|super high-jump]] cardinal, [[high-jump|high-jump with unbounded excess closure]] cardinal
* [[Vopenka | Vopěnka's principle]], [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal
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* [[Woodin#Shelah cardinals|Shelah for supercompactness]]
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* [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin#Shelah cardinals|Woodin for supercompactness]] cardinal
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* [[Vopenka | Vopěnka's principle]]
 
* [[extendible]] cardinal, [[extendible | $\alpha$-extendible]] cardinal
 
* [[extendible]] cardinal, [[extendible | $\alpha$-extendible]] cardinal
* [[grand reflection]] cardinal
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<!--* [[grand reflection]] cardinal-->
 
* [[hypercompact]] cardinal
 
* [[hypercompact]] cardinal
* '''[[supercompact]]''' cardinal, [[supercompact | $\lambda$-supercompact]] cardinal, [[PFA]] cardinal
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* '''[[supercompact]]''' cardinal, [[supercompact | $\lambda$-supercompact]] cardinal
* '''[[strongly compact]]''' cardinal
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* '''[[strongly compact]]''' cardinal  [[strongly compact | $\lambda$-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
* [[indestructible weakly compact]] cardinal
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* [[Weakly_compact#Indestructibility of a weakly compact cardinal|indestructible weakly compact]] cardinal
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* The '''[[proper forcing axiom]]''' and [[forcing#Proper forcing|Martin's maximum]]
 
* [[subcompact]] cardinal
 
* [[subcompact]] cardinal
* '''[[superstrong]]''' cardinal
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* [[superstrong]] cardinal
* [[Shelah]] cardinal
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* [[Woodin#Shelah|Shelah]] cardinal
* The '''[[axiom of determinacy]]'''
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* The '''[[axiom of determinacy]]''' and [[axiom of projective determinacy|its projective counterpart]]
 
* '''[[Woodin]]''' cardinal
 
* '''[[Woodin]]''' cardinal
* '''[[strong]]''' cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy
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* [[strongly tall]] cardinal
* [[tall]] cardinal
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* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchies, [[tall]] cardinal, [[tall|$\theta$-tall]] hierarchy
* [[zero dagger| $0^\dagger$]], $j:L[U]\to L[U]$ cardinal
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* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]]  
 
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]]  
* '''[[measurable]]''' cardinal, [[weakly measurable]] cardinal
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*[[zero dagger| $0^\dagger$]] (''zero-dagger'')
* '''[[Ramsey]]''' cardinal, [[strongly Ramsey]] cardinal, [[virtually Ramsey]] cardinal
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* '''[[measurable]]''' cardinal, [[weakly measurable]] cardinal, singular [[Jonsson|Jónsson]] cardinal
* [[Rowbottom]] cardinal
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* [[Ramsey#Super Ramsey cardinal|super Ramsey]] cardinal
* [[Jonsson | Jónsson]] cardinal
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* [[Ramsey#Strongly Ramsey cardinal|strongly Ramsey]] cardinal
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$  
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* '''[[Ramsey]]''' cardinal, [[Jonsson | Jónsson]] cardinal, [[Rowbottom]] cardinal, [[Ramsey#Virtually Ramsey cardinal|virtually Ramsey]] cardinal
* '''[[zero sharp | $0^\sharp$]]''', $j:L\to L$ cardinal
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* [[Ramsey#Almost Ramsey cardinal|almost Ramsey]] cardinal
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* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$, [[Chang's conjecture]]
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* [[Ramsey#.24.5Calpha.24-iterable cardinal|$\omega_1$-iterable]] cardinal
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* '''[[zero sharp | $0^\sharp$]]''' (''zero-sharp''), existence of [[Constructible universe#Silver indiscernibles|Silver indiscernibles]]
 
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$
 
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$
* [[Ramsey | $1$-iterable]] cardinal, and the [[Ramsey | $\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$
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* the [[Ramsey#.24.5Calpha.24-iterable cardinal| $\alpha$-iterable]] cardinals hierarchy for $1\leq\alpha<\omega_1$
 
* [[remarkable]] cardinal
 
* [[remarkable]] cardinal
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* [[Ramsey#.24.5Calpha.24-iterable cardinal|weakly Ramsey]] cardinal
 
* [[ineffable]] cardinal, [[weakly ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy; [[completely ineffable]] cardinal
 
* [[ineffable]] cardinal, [[weakly ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy; [[completely ineffable]] cardinal
 
* [[subtle]] cardinal
 
* [[subtle]] cardinal
* [[ethereal]] cardinal
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* [[ineffable#Ethereal cardinal|ethereal]] cardinal
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, [[uplifting#strongly uplifting | strongly uplifting]] cardinal  
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* [[unfoldable#Superstrongly Unfoldable | superstrongly unfoldable]] cardinal, [[uplifting#strongly uplifting | strongly uplifting]] cardinal  
 
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal
 
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal
* [[unfoldable]] cardinal, [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal
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* [[unfoldable]] cardinal, [[unfoldable#Strongly Unfoldable | strongly unfoldable]] cardinal
* [[indescribable]] cardinal, [[totally indescribable]] cardinal
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* [[indescribable]] hierarchy, [[totally indescribable]] cardinal
 
* '''[[weakly compact]]''' cardinal
 
* '''[[weakly compact]]''' cardinal
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* The [[Positive set theory|positive set theory]] $\text{GPK}^+_\infty$
 
* '''[[Mahlo]]''' cardinal, [[Mahlo#Hyper-Mahlo | $1$-Mahlo]], the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy, [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals
 
* '''[[Mahlo]]''' cardinal, [[Mahlo#Hyper-Mahlo | $1$-Mahlo]], the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy, [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal, [[uplifting]] cardinal
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* [[uplifting]] cardinal, [[uplifting#pseudo uplifting cardinal | pseudo uplifting]] cardinal
* [[ORD is Mahlo]]
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* [[ORD is Mahlo|$\text{Ord}$ is Mahlo]]
 
* [[reflecting#Sigma_2 correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals
 
* [[reflecting#Sigma_2 correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals
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* [[Jäger's collapsing functions and ρ-inaccessible ordinals]]
 
* [[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
* (strongly) '''[[inaccessible]]''' cardinal, '''[[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]]''' cardinal
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* '''[[inaccessible]]''' cardinal, '''[[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]]''' cardinal
* [[Kelly-Morse]] set theory
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* [[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
 
* '''[[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]]  
 
* the [[Transitive ZFC model#Transitive model universe axiom | transitive model universe axiom]]  
* [[Transitive ZFC model]]
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* [[transitive ZFC model|transitive model of $\text{ZFC}$]]
 
* 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#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]]  
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* '''[[Con ZFC | $\text{Con(ZFC)}$]]''' and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]]
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* '''[[ZFC|Zermelo-Fraenkel]]''' set theory
  
 
* down to [[the middle attic]]
 
* down to [[the middle attic]]

Revision as of 08:49, 7 February 2019

Cape Pogue Lighthouse photo by Timothy Valentine

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.