Difference between revisions of "Upper attic"

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(ineffable (=normal $0$-Ramsey)!)
(Bagaria 2002)
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* [[shrewd|$η$-shrewd]], [[shrewd]], [[shrewd|$\mathcal{A}$-$η$-shrewd]], [[shrewd|$\mathcal{A}$-shrewd]] cardinals
 
* [[shrewd|$η$-shrewd]], [[shrewd]], [[shrewd|$\mathcal{A}$-$η$-shrewd]], [[shrewd|$\mathcal{A}$-shrewd]] cardinals
 
* [[indescribable]] hierarchy, [[totally indescribable]] cardinal
 
* [[indescribable]] hierarchy, [[totally indescribable]] cardinal
* '''[[weakly compact]]''' (=$\Pi_1^1$-[[indescribable]]=$0$-[[Ramsey]]) cardinal
+
* [[weakly compact|$\Sigma_n$-weakly compact]] cardinals, [[weakly compact|$\Sigma_\omega$-weakly compact]] cardinal, '''[[weakly compact]]''' (=$\Pi_1^1$-[[indescribable]]=$0$-[[Ramsey]]) cardinal
 
* The [[Positive set theory|positive set theory]] $\text{GPK}^+_\infty$  
 
* The [[Positive set theory|positive set theory]] $\text{GPK}^+_\infty$  
* (strongly) '''[[Mahlo]]''' cardinal, [[Mahlo|weakly Mahlo]] cardinal, [[Mahlo#Hyper-Mahlo|$1$-Mahlo]], the [[Mahlo#Hyper-Mahlo|$\alpha$-Mahlo]] hierarchy, [[Mahlo#Hyper-Mahlo|hyper-Mahlo]] cardinals, [[Mahlo|$Ω^α$-Mahlo]] cardinals
+
* [[Mahlo|$\Sigma_\omega$-Mahlo]] cardinal, [[Mahlo|weakly Mahlo]] cardinal, (strongly) '''[[Mahlo]]''' cardinal, [[Mahlo#Hyper-Mahlo|$1$-Mahlo]], the [[Mahlo#Hyper-Mahlo|$\alpha$-Mahlo]] hierarchy, [[Mahlo#Hyper-Mahlo|hyper-Mahlo]] cardinals, [[Mahlo|$Ω^α$-Mahlo]] cardinals
 
* [[uplifting#pseudo uplifting cardinal|pseudo uplifting]] cardinal, [[uplifting]] cardinal
 
* [[uplifting#pseudo uplifting cardinal|pseudo uplifting]] cardinal, [[uplifting]] cardinal
 
* [[ORD is Mahlo|$\text{Ord}$ is Mahlo]]
 
* [[ORD is Mahlo|$\text{Ord}$ is Mahlo]]
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* [[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#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]] (the existence of a proper class of [[inaccessible]] cardinals)
 
* [[inaccessible#Universes|Grothendieck universe axiom]] (the existence of a proper class of [[inaccessible]] cardinals)
* (strongly) '''[[inaccessible]]''' cardinal, [[inaccessible#Weakly inaccessible cardinal|weakly inaccessible]] cardinal
+
* [[inaccessible#Weakly inaccessible cardinal|weakly inaccessible]] cardinal, (strongly) '''[[inaccessible]]''' cardinal
 
* [[Morse-Kelley set theory|Morse-Kelley]] set theory
 
* [[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

Revision as of 10:27, 6 September 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.