Difference between revisions of "Upper attic"

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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.  
  
* [[hypercompact|excessively hypercompact cardinal]] (inconsistent)
+
* [[hypercompact|excessively hypercompact cardinal]]
* 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 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]
+
* '''[[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 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]
* The [[wholeness axiom]]
+
* The '''[[wholeness axiom]]'''
* [[huge | almost n-huge]] cardinal, [[huge|n-huge]] cardinal, [[superstrong|(n+1)-superstrong cardinal]], [[huge|super-n-huge]] cardinal
+
* '''[[huge|n-huge]]''' cardinal, [[huge | almost n-huge]] cardinal, [[huge|super-n-huge]] cardinal, [[superstrong|(n+1)-superstrong cardinal]]
* [[Vopenka | Vopěnka's principle]], [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal, [[high-jump]] 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
* [[extendible | $\alpha$-extendible]] cardinal, [[extendible]] cardinal
+
* '''[[Vopenka | Vopěnka's principle]]''', [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal
 +
* '''[[extendible]] cardinal''', [[extendible | $\alpha$-extendible]] cardinal
 
* [[grand reflection]] cardinal
 
* [[grand reflection]] cardinal
 
* [[hypercompact]] cardinal
 
* [[hypercompact]] cardinal
* [[supercompact | $\lambda$-supercompact]] cardinal, [[supercompact]] cardinal
+
* '''[[supercompact]] cardinal''', [[supercompact | $\lambda$-supercompact]] cardinal
 
* [[PFA]] cardinal
 
* [[PFA]] cardinal
* [[strongly compact]] cardinal
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* '''[[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
 
* [[indestructible weakly compact]] cardinal
 
* [[subcompact]] cardinal
 
* [[subcompact]] cardinal
* [[superstrong]] cardinal
+
* '''[[superstrong]] cardinal'''
 
* [[Shelah]] cardinal
 
* [[Shelah]] cardinal
* [[Woodin]] cardinal, the [[axiom of determinacy]]
+
* '''[[Woodin]] cardinal''', the [[axiom of determinacy]]
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy
+
* '''[[strong]] cardinal''' and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy
 
* [[tall]] cardinal
 
* [[tall]] cardinal
 
* [[zero dagger| $0^\dagger$]], $j:L[U]\to L[U]$ cardinal
 
* [[zero dagger| $0^\dagger$]], $j:L[U]\to L[U]$ cardinal
 
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]]  
 
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]]  
* [[weakly measurable]] cardinal, [[measurable]] cardinal
+
* '''[[measurable]] cardinal''', [[weakly measurable]] cardinal
 
* [[virtually Ramsey]] cardinal, [[Ramsey]] cardinal, [[strongly Ramsey]] cardinal
 
* [[virtually Ramsey]] cardinal, [[Ramsey]] cardinal, [[strongly Ramsey]] cardinal
 
* [[Rowbottom]] cardinal
 
* [[Rowbottom]] cardinal
 
* [[Jonsson | Jónsson]] cardinal
 
* [[Jonsson | Jónsson]] cardinal
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$  
+
* '''[[Erdos | $\omega_1$-Erdős]] cardinal''' and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$  
* [[zero sharp | $0^\sharp$]], $j:L\to L$ cardinal
+
* '''[[zero sharp | $0^\sharp$]]''', $j:L\to L$ cardinal
 
* [[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 $1\leq \alpha\leq \omega_1$
 
* [[Ramsey | $1$-iterable]] cardinal, and the [[Ramsey | $\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$
 
* [[remarkable]] cardinal
 
* [[remarkable]] cardinal
* [[weakly ineffable]] cardinal, [[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
 
* [[ethereal]] cardinal
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* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal
 
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal
 
* [[unfoldable]] cardinal, [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal
 
* [[unfoldable]] cardinal, [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal
* [[indescribable]] cardinal, [[totally indescribable]] cardinal
+
* '''[[indescribable]] cardinal''', [[totally indescribable]] cardinal
* [[weakly compact]] cardinal
+
* '''[[weakly compact]] cardinal'''
* [[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
* [[Mahlo]] cardinal, [[Mahlo#Hyper-Mahlo | $1$-Mahlo]], the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy  
+
 
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal, [[uplifting]] cardinal
 
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal, [[uplifting]] cardinal
 
* [[ORD is Mahlo]]
 
* [[ORD is Mahlo]]
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* [[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
* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal, (strongly) [[inaccessible]] cardinal,
+
* '''[[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal''', (strongly) '''[[inaccessible]] cardinal''',
 
* [[Kelly-Morse]] set theory
 
* [[Kelly-Morse]] 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]]
 
* [[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#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]]  
+
* '''[[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 06:15, 13 September 2017

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.