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.  
  
* The [[Kunen inconsistency]]
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* The [[Kunen inconsistency]]: [[Reinhardt]] cardinal, [[Kunen_inconsistency#Super_Reinhardt_cardinal | super Reinhardt]] cardinal, [[Berkeley]] cardinal
* [[Berkeley]] cardinal
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* [[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 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]
* [[Kunen_inconsistency#Super_Reinhardt_cardinal | super Reinhardt]] cardinal, [[Reinhardt]] cardinal
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* [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]
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* [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$  
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* [[rank into rank]] cardinal $j:V_\lambda\to V_\lambda$
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* The [[wholeness axiom]]
 
* The [[wholeness axiom]]
* [[huge#Super_n-huge | super $n$-huge]] cardinal
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* [[huge | almost n-huge]] cardinal, [[huge|n-huge]] cardinal, [[superstrong|(n+1)-superstrong cardinal]], [[huge|super-n-huge]] cardinal
* [[huge#Superhuge | superhuge]] cardinal
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* [[Vopenka | Vopěnka's principle]], [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal, [[high-jump]] cardinal
* [[huge]] cardinal
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* [[extendible | $\alpha$-extendible]] cardinal, [[extendible]] cardinal
* [[huge#Almost huge | almost huge]] cardinal
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* [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal
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* [[Vopenka | Vopěnka's principle]]
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* [[extendible]] cardinal
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* [[grand reflection]] cardinal
 
* [[grand reflection]] cardinal
* [[supercompact]] cardinal
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* [[supercompact | $\lambda$-supercompact]] cardinal, [[supercompact]] cardinal
 
* [[PFA]] cardinal
 
* [[PFA]] cardinal
 
* [[strongly compact]] cardinal
 
* [[strongly compact]] cardinal
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* [[superstrong]] cardinal
 
* [[superstrong]] cardinal
 
* [[Shelah]] cardinal
 
* [[Shelah]] cardinal
* [[Woodin]] cardinal
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* [[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| $0^\dagger$]]
 
* [[zero dagger| $0^\dagger$]]
 
* 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
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* [[weakly measurable]] cardinal, [[measurable]] cardinal
* [[weakly measurable]] cardinal
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* [[virtually Ramsey]] cardinal, [[Ramsey]] cardinal, [[strongly Ramsey]] cardinal
* [[strongly Ramsey]] cardinal
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* [[Ramsey]] cardinal
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* [[virtually Ramsey]] cardinal
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* [[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 | $0^\sharp$]]
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* [[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
* [[completely ineffable]] cardinal
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* [[weakly ineffable]] cardinal, [[ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy; [[completely ineffable]] cardinal
* [[ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy
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* [[weakly ineffable]] cardinal
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* [[subtle]] cardinal
 
* [[subtle]] cardinal
 
* [[ethereal]] cardinal
 
* [[ethereal]] cardinal
 
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, [[uplifting#strongly uplifting | strongly uplifting]] cardinal  
 
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, [[uplifting#strongly uplifting | strongly uplifting]] cardinal  
 
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal
 
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal
* [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal
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* [[unfoldable]] cardinal, [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal
* [[unfoldable]] cardinal
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* [[indescribable]] cardinal, [[totally indescribable]] cardinal
* [[Totally indescribable]] cardinal
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* [[indescribable]] cardinal
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* [[weakly compact]] cardinal
 
* [[weakly compact]] cardinal
 
* [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals  
 
* [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals  
* the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy
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* [[Mahlo]] cardinal, [[Mahlo#Hyper-Mahlo | $1$-Mahlo]], the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy
* [[Mahlo#Hyper-Mahlo | $1$-Mahlo]]
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* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal, [[uplifting]] cardinal
* [[Mahlo]] cardinal
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* [[uplifting]] cardinal
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* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal
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* [[ORD is Mahlo]]
 
* [[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
 
* [[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]] cardinal, also known as strongly inaccessible
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* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal, (strongly) [[inaccessible]] cardinal,
* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal
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* [[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

Revision as of 05:26, 10 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.