Difference between revisions of "Upper attic"

From Cantor's Attic
Jump to: navigation, search
(Bagaria 2002)
m ($\Sigma_n$-Mahlo)
Line 68: Line 68:
 
* [[weakly compact|$\Sigma_n$-weakly compact]] cardinals, [[weakly compact|$\Sigma_\omega$-weakly compact]] 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]] $\text{GPK}^+_\infty$  
* [[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
+
* [[Mahlo|$\Sigma_n$-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]]

Revision as of 11:57, 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.