Difference between revisions of "Upper attic"

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(unfoldable are weaker but equiconsistent)
(this guesses seem to be closer)
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* $\Sigma^m_n$- and '''$\Pi^m_n$-[[indescribable]]''', [[totally indescribable]], [[indescribable|$η$-indescribable]] cardinals
 
* $\Sigma^m_n$- and '''$\Pi^m_n$-[[indescribable]]''', [[totally indescribable]], [[indescribable|$η$-indescribable]] cardinals
 
* uncountable cardinal with the [[tree property]], '''[[weakly compact]]''' (=$\Pi_1^1$-[[indescribable]]=$0$-[[Ramsey]]) cardinal
 
* uncountable cardinal with the [[tree property]], '''[[weakly compact]]''' (=$\Pi_1^1$-[[indescribable]]=$0$-[[Ramsey]]) cardinal
* [[weakly compact|$\Sigma_n$-weakly compact]] cardinals, [[weakly compact|$\Sigma_\omega$-weakly compact]] cardinal
+
* The [[Positive set theory|positive set theory]] $\text{GPK}^+_\infty$
* The [[Positive set theory|positive set theory]] $\text{GPK}^+_\infty$  
+
* [[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
+
* [[Mahlo|weakly Mahlo]] cardinal, (strongly) '''[[Mahlo]]''' cardinal
 
* [[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]]<!-- apparently equiconsistent with a reflecting cardinal -->
 
* [[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
 +
* [[Mahlo|$\Sigma_n$-Mahlo]], [[weakly compact|$\Sigma_n$-weakly compact]], [[Mahlo|$\Sigma_\omega$-Mahlo]] and [[weakly compact|$\Sigma_\omega$-weakly compact]]<!-- Really? In particular, are $\Sigma_\omega$ variants not stronger then ORD is Mahlo? Maybe $\Sigma_\omega$-weakly compact is even stronger than Mahlo? --> cardinals
 
* [[Jäger's collapsing functions and ρ-inaccessible ordinals]]  
 
* [[Jäger's collapsing functions and ρ-inaccessible ordinals]]  
 
* [[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

Revision as of 13:48, 21 April 2022

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.