Difference between revisions of "Upper attic"

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(Misrepresents the other of large cardinals. Hyperhuge=2-Supercompact>2-Strong)
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* [[Reinhardt|weakly Reinhardt]], [[Reinhardt]], [[Reinhardt|super Reinhardt]], [[Reinhardt|totally Reinhardt]] cardinal
 
* [[Reinhardt|weakly Reinhardt]], [[Reinhardt]], [[Reinhardt|super Reinhardt]], [[Reinhardt|totally Reinhardt]] cardinal
 
* the '''[[Kunen inconsistency]]'''
 
* the '''[[Kunen inconsistency]]'''
* '''[[rank into rank]]''' axioms, $C^{(n)}$-I3, $C^{(n)}$-I1, [[L of V_lambda+1|I0 axiom]] and strengthenings
+
* '''[[rank into rank]]''' axioms, $C^{(n)}$-I3, $C^{(n)}$-I1, [[n-fold variants|$\omega$-fold variants]], [[L of V_lambda+1|I0 axiom]] and strengthenings
 
* The [[wholeness axioms]], [[I4|axioms $\mathrm{I}_4^n$]]
 
* The [[wholeness axioms]], [[I4|axioms $\mathrm{I}_4^n$]]
* [[n-fold variants]] of hugeness (plus $C^{(n)}$ variants), extendibility, supercompactness, strongness, etc...
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* [[n-fold variants|$n$-fold variants]] of hugeness (plus $C^{(n)}$ variants), extendibility, supercompactness, strongness, etc...
 
* [[huge|almost huge]], '''[[huge]]''', [[huge|huge*]], [[huge|super almost huge]], [[huge|superhuge]], [[huge|ultrahuge]], [[superstrong|2-superstrong]] cardinal
 
* [[huge|almost huge]], '''[[huge]]''', [[huge|huge*]], [[huge|super almost huge]], [[huge|superhuge]], [[huge|ultrahuge]], [[superstrong|2-superstrong]] cardinal
 
* [[high-jump]] cardinal, [[high-jump|almost high-jump]] cardinal, [[high-jump|super high-jump]] cardinal, [[high-jump|high-jump with unbounded excess closure]] cardinal
 
* [[high-jump]] cardinal, [[high-jump|almost high-jump]] cardinal, [[high-jump|super high-jump]] cardinal, [[high-jump|high-jump with unbounded excess closure]] cardinal
 
* [[Woodin#Shelah cardinals|Shelah for supercompactness]]
 
* [[Woodin#Shelah cardinals|Shelah for supercompactness]]
* [[Vopenka#Vopěnka cardinals|Vopěnka]] (=[[Woodin#Shelah cardinals|Woodin for supercompactness]]) cardinal
+
* [[Vopenka|Vopěnka scheme]], '''[[Vopenka|Vopěnka principle]]''', [[Vopenka#Vopěnka cardinals|Vopěnka-scheme]] cardinal, [[Vopenka#Vopěnka cardinals|Vopěnka]] (=[[Woodin#Shelah cardinals|Woodin for supercompactness]]) cardinal
* [[Vopenka|Vopěnka's principle]]
+
* [[extendible|$\alpha$-extendible]] hierarchy, '''[[extendible]]''' cardinal, [[extendible|$C^{(n)}$-extendible]] hierarchy
* [[extendible|$C^{(n)}$-extendible]] hierarchy
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* [[extendible]] cardinal, [[extendible|$\alpha$-extendible]] cardinal
+
 
* [[Woodin|Woodin for strong compactness]]
 
* [[Woodin|Woodin for strong compactness]]
 
<!--* [[grand reflection]] cardinal-->
 
<!--* [[grand reflection]] cardinal-->
* [[hypercompact]] cardinal
+
* [[Supercompact#Enhanced supercompact cardinals|enhanced $\lambda$-supercompact]] cardinals, [[Supercompact#Enhanced supercompact cardinals|enhanced supercompact]] cardinal, [[hypercompact|$\lambda$-hypercompact]] cardinals, [[hypercompact]] cardinal
* '''[[supercompact]]''' cardinal, [[supercompact|$\lambda$-supercompact]] cardinal
+
* [[supercompact|$\lambda$-supercompact]] cardinals, '''[[supercompact]]''' cardinal
* '''[[strongly compact]]''' cardinal [[strongly compact|$\lambda$-strongly compact]] cardinal
+
* [[strongly compact|$\lambda$-strongly compact]] cardinals, '''[[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
 
* [[Weakly_compact#Indestructibility of a weakly compact cardinal|indestructible weakly compact]] cardinal
 
* [[Weakly_compact#Indestructibility of a weakly compact cardinal|indestructible weakly compact]] cardinal
 
* The '''[[proper forcing axiom]]''' and [[forcing#Proper forcing|Martin's maximum]]
 
* The '''[[proper forcing axiom]]''' and [[forcing#Proper forcing|Martin's maximum]]
 
* [[subcompact]] cardinal
 
* [[subcompact]] cardinal
* [[superstrong|$C^{(n)}$-superstrong]] hierarchy
+
* [[superstrong]] cardinal, [[superstrong|$C^{(n)}$-superstrong]] hierarchy
* [[superstrong]] cardinal
+
 
* [[Woodin#Shelah|Shelah]] cardinal
 
* [[Woodin#Shelah|Shelah]] cardinal
 
* The '''[[axiom of determinacy]]''' and [[axiom of projective determinacy|its projective counterpart]]
 
* The '''[[axiom of determinacy]]''' and [[axiom of projective determinacy|its projective counterpart]]
 
* '''[[Woodin]]''' cardinal
 
* '''[[Woodin]]''' cardinal
 
* [[strongly tall]] cardinal
 
* [[strongly tall]] cardinal
* [[strong]] cardinal and the [[strong|$\theta$-strong]] and [[strong#Hypermeasurable|hypermeasurability]] hierarchies, [[tall]] cardinal, [[tall|$\theta$-tall]] hierarchy
+
* the [[strong|$\theta$-strong]], [[strong#Hypermeasurable|hypermeasurability]] and [[tall|$\theta$-tall]] hierarchies, [[tall]] and '''[[strong]]''' cardinals
 
* 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| $0^\dagger$]] (''zero-dagger'')
 
*[[zero dagger| $0^\dagger$]] (''zero-dagger'')
* '''[[measurable]]''' cardinal, [[weakly measurable]] cardinal, singular [[Jonsson|Jónsson]] cardinal, $κ^+$-[[filter property]], [[Ramsey|strategic $(\omega+1)$-Ramsey]] cardinal, [[Ramsey|strategic fully Ramsey]] cardinal, [[Ramsey|$ω_1$-very Ramsey]] cardinal, [[Ramsey|$κ$-very Ramsey]] cardinal
+
* [[weakly measurable]] cardinal, '''[[measurable]]''' cardinal
 +
** singular [[Jonsson|Jónsson]] cardinal
 +
** $κ^+$-[[filter property]], [[Ramsey|strategic $(\omega+1)$-Ramsey]] cardinal, [[Ramsey|strategic fully Ramsey]] cardinal, [[Ramsey|$ω_1$-very Ramsey]] cardinal, [[Ramsey|$κ$-very Ramsey]] cardinal
 
* $κ$-[[filter property]], [[Ramsey|fully Ramsey]] (=[[Ramsey|$κ$-Ramsey]]) cardinal  
 
* $κ$-[[filter property]], [[Ramsey|fully Ramsey]] (=[[Ramsey|$κ$-Ramsey]]) cardinal  
 
* [[Ramsey#Strongly Ramsey cardinal|strongly Ramsey]] cardinal, [[Ramsey|strongly Ramsey M-rank]], [[Ramsey#Super Ramsey cardinal|super Ramsey]] cardinal, [[Ramsey|super Ramsey M-rank]]
 
* [[Ramsey#Strongly Ramsey cardinal|strongly Ramsey]] cardinal, [[Ramsey|strongly Ramsey M-rank]], [[Ramsey#Super Ramsey cardinal|super Ramsey]] cardinal, [[Ramsey|super Ramsey M-rank]]
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* [[Ramsey|$\alpha$-Mahlo–Ramsey]] hierarchy
 
* [[Ramsey|$\alpha$-Mahlo–Ramsey]] hierarchy
 
* [[Ramsey|Ramsey M-rank]]
 
* [[Ramsey|Ramsey M-rank]]
* '''[[Ramsey]]''' cardinal, [[Jonsson|Jónsson]] cardinal, [[Rowbottom]] cardinal, [[Ramsey#Virtually Ramsey cardinal|virtually Ramsey]] cardinal
+
* [[Ramsey#Virtually Ramsey cardinal|virtually Ramsey]] cardinal, [[Jonsson|Jónsson]] cardinal, [[Rowbottom]] cardinal, '''[[Ramsey]]''' cardinal
 
* [[Erdos|$\alpha$-weakly Erdős]] cardinals, [[Erdos|greatly Erdős]] cardinal
 
* [[Erdos|$\alpha$-weakly Erdős]] cardinals, [[Erdos|greatly Erdős]] cardinal
 
* [[Ramsey#Almost Ramsey cardinal|almost Ramsey]] cardinal
 
* [[Ramsey#Almost Ramsey cardinal|almost Ramsey]] cardinal
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* '''[[zero sharp|$0^\sharp$]] (''zero-sharp'')''', existence of [[Constructible universe#Silver indiscernibles|Silver indiscernibles]]
 
* '''[[zero sharp|$0^\sharp$]] (''zero-sharp'')''', existence of [[Constructible universe#Silver indiscernibles|Silver indiscernibles]]
 
* [[Silver cardinal]]
 
* [[Silver cardinal]]
* the [[Erdos|$\alpha$-Erdős]], [[Ramsey#.24.5Calpha.24-iterable cardinal|$\alpha$-iterable]] and [[Ramsey|$(\omega, \alpha)$-Ramsey]] hierarchy for countable infinite $\alpha$
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* the [[Erdos|$\alpha$-'''Erdős''']], [[Ramsey#.24.5Calpha.24-iterable cardinal|$\alpha$-iterable]] and [[Ramsey|$(\omega, \alpha)$-Ramsey]] hierarchy for countable infinite $\alpha$
 
* [[Erdos|$\omega$-Erdős]] cardinal, [[remarkable|weakly remarkable]] cardinal that is not remarkable
 
* [[Erdos|$\omega$-Erdős]] cardinal, [[remarkable|weakly remarkable]] cardinal that is not remarkable
 
* [[rank into rank|virtually rank-into-rank]] cardinal
 
* [[rank into rank|virtually rank-into-rank]] cardinal
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* [[Woodin|virtually Shelah for supercompactness]] cardinal
 
* [[Woodin|virtually Shelah for supercompactness]] cardinal
 
* [[extendible|virtually extendible]] (=[[remarkable|$2$-remarkable]]), [[extendible|virtually $C^{(n)}$-extendible]] (=[[remarkable|$n+1$-remarkable]]) cardinals, [[remarkable|completely remarkable]] cardinal, [[Vopenka|Generic Vopěnka's Principle]]
 
* [[extendible|virtually extendible]] (=[[remarkable|$2$-remarkable]]), [[extendible|virtually $C^{(n)}$-extendible]] (=[[remarkable|$n+1$-remarkable]]) cardinals, [[remarkable|completely remarkable]] cardinal, [[Vopenka|Generic Vopěnka's Principle]]
* [[remarkable|($1$-)remarkable]] (=virtually supercompact), [[measurable|virtually measurable]], [[Ramsey|strategic $\omega$-Ramsey]] cardinals, [[proper forcing axiom|weak Proper Forcing Axiom]]
+
* [[remarkable|($1$-)'''remarkable''']] (=virtually supercompact), [[measurable|virtually measurable]], [[Ramsey|strategic $\omega$-Ramsey]] cardinals, [[proper forcing axiom|weak Proper Forcing Axiom]]
 
* [[Ramsey#.24.5Calpha.24-iterable cardinal|weakly Ramsey]] (=$1$-iterable) cardinal, [[Ramsey|super weakly Ramsey]] cardinals, [[Ramsey|$\omega$-Ramsey]] cardinal
 
* [[Ramsey#.24.5Calpha.24-iterable cardinal|weakly Ramsey]] (=$1$-iterable) cardinal, [[Ramsey|super weakly Ramsey]] cardinals, [[Ramsey|$\omega$-Ramsey]] cardinal
 
* [[completely ineffable]] cardinal (= $\omega$-[[filter property]])
 
* [[completely ineffable]] cardinal (= $\omega$-[[filter property]])
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* [[subtle]] cardinal
 
* [[subtle]] cardinal
 
* [[ineffable#Ethereal cardinal|ethereal]] cardinal
 
* [[ineffable#Ethereal cardinal|ethereal]] cardinal
* [[uplifting#Strongly Uplifting|strongly uplifting]] cardinal (=[[unfoldable#Superstrongly Unfoldable|superstrongly unfoldable]])
+
* [[uplifting#Strongly Uplifting|strongly uplifting]] (=[[unfoldable#Superstrongly Unfoldable|superstrongly unfoldable]]) cardinal
 
* [[weakly superstrong]] cardinal
 
* [[weakly superstrong]] cardinal
 
* [[unfoldable]] cardinal, [[unfoldable#Strongly Unfoldable|strongly unfoldable]] cardinal
 
* [[unfoldable]] cardinal, [[unfoldable#Strongly Unfoldable|strongly unfoldable]] cardinal
 
* [[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
+
* $\Sigma^m_n$- and '''$\Pi^m_n$-[[indescribable]]''', [[totally indescribable]], [[indescribable|$η$-indescribable]] cardinals
 
* [[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]] $\text{GPK}^+_\infty$  

Revision as of 06:10, 19 October 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.