Difference between revisions of "Upper attic"

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* 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 | $j:\text{L}(V_{\lambda+1})\to\text{L}(V_{\lambda+1})$]]
 
* The [[wholeness axioms]]
 
* The [[wholeness axioms]]
 
* [[n-fold variants | n-fold supercompact]], [[n-fold variants | n-fold strong]], [[n-fold variants | n-fold extendible]], [[n-fold variants | n-fold Woodin]]
 
* [[n-fold variants | n-fold supercompact]], [[n-fold variants | n-fold strong]], [[n-fold variants | n-fold extendible]], [[n-fold variants | n-fold Woodin]]
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<!--* [[grand reflection]] cardinal-->
 
<!--* [[grand reflection]] cardinal-->
 
* [[hypercompact]] cardinal
 
* [[hypercompact]] cardinal
* '''[[supercompact]]''' cardinal, [[supercompact | $\lambda$-supercompact]] cardinal, [[PFA]] cardinal
+
* '''[[supercompact]]''' cardinal, [[supercompact | $\lambda$-supercompact]] cardinal, [[PFA|$\text{PFA}$]] cardinal
 
* '''[[strongly compact]]''' cardinal
 
* '''[[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
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* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchies
 
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchies
 
* [[tall]] cardinal
 
* [[tall]] 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 | $o(\kappa)=\kappa^{++}$]]  
 +
* [[zero dagger| $0^\dagger$]], $j:\text{L}[U]\to\text{L}[U]$ cardinal
 
* '''[[measurable]]''' cardinal, [[weakly measurable]] cardinal
 
* '''[[measurable]]''' cardinal, [[weakly measurable]] cardinal
 
* '''[[Ramsey]]''' cardinal, [[strongly Ramsey]] cardinal, [[virtually Ramsey]] cardinal
 
* '''[[Ramsey]]''' cardinal, [[strongly Ramsey]] cardinal, [[virtually Ramsey]] cardinal
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* [[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 | $0^\sharp$]]''', $j:\text{L}\to\text{L}$ cardinal
 
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$
 
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$
 
* the [[Ramsey#.24.5Calpha.24-iterable cardinal| $\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$
 
* the [[Ramsey#.24.5Calpha.24-iterable cardinal| $\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$
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* [[indescribable]] cardinal, [[totally indescribable]] cardinal
 
* [[indescribable]] cardinal, [[totally indescribable]] cardinal
 
* '''[[weakly compact]]''' cardinal
 
* '''[[weakly compact]]''' cardinal
* The [[Positive set theory|positive set theory]] $GPK^+_\infty$  
+
* 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]] cardinal, [[uplifting#pseudo uplifting cardinal | pseudo uplifting]] cardinal
 
* [[uplifting]] cardinal, [[uplifting#pseudo uplifting cardinal | pseudo uplifting]] cardinal
* [[ORD is Mahlo]]
+
* [[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
 
* [[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
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* '''[[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|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]]
+
* '''[[Con ZFC | $\text{Con(ZFC)}$]]''' and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]]
 
* '''[[ZFC|Zermelo-Fraenkel]]''' set theory
 
* '''[[ZFC|Zermelo-Fraenkel]]''' set theory
  
 
* down to [[the middle attic]]
 
* down to [[the middle attic]]

Revision as of 14:10, 11 November 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.