Difference between revisions of "Upper attic"

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* '''[[huge]]''' cardinal, [[huge|superhuge]] cardinal, [[huge|ultrahuge]] cardinal, [[superstrong|2-superstrong]] cardinal
 
* '''[[huge]]''' cardinal, [[huge|superhuge]] cardinal, [[huge|ultrahuge]] cardinal, [[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
* [[Vopenka | Vopěnka's principle]], [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin|Woodin for supercompactness]] cardinal
+
* [[Woodin#Shelah cardinals|Shelah for supercompactness]]
 +
* [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin#Shelah cardinals|Woodin for supercompactness]] cardinal
 +
* [[Vopenka | Vopěnka's principle]]
 
* [[extendible]] cardinal, [[extendible | $\alpha$-extendible]] cardinal
 
* [[extendible]] cardinal, [[extendible | $\alpha$-extendible]] cardinal
 
<!--* [[grand reflection]] cardinal-->
 
<!--* [[grand reflection]] cardinal-->
 
* [[hypercompact]] cardinal
 
* [[hypercompact]] cardinal
 
* '''[[supercompact]]''' cardinal, [[supercompact | $\lambda$-supercompact]] cardinal
 
* '''[[supercompact]]''' cardinal, [[supercompact | $\lambda$-supercompact]] cardinal
* '''[[strongly compact]]''' cardinal
+
* '''[[strongly compact]]''' cardinal  [[strongly compact | $\lambda$-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
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* [[subcompact]] cardinal
 
* [[subcompact]] cardinal
 
* [[superstrong]] cardinal
 
* [[superstrong]] cardinal
* [[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
 
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchies, [[tall]] cardinal, [[tall|$\theta$-tall]] hierarchy
 
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchies, [[tall]] cardinal, [[tall|$\theta$-tall]] hierarchy
 
* 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| $0^\dagger$]] (''zero-dagger'')
 
* '''[[measurable]]''' cardinal, [[weakly measurable]] cardinal, singular [[Jonsson|Jónsson]] cardinal
 
* '''[[measurable]]''' cardinal, [[weakly measurable]] cardinal, singular [[Jonsson|Jónsson]] cardinal
* '''[[Ramsey]]''' cardinal, [[Jonsson | Jónsson]] cardinal, [[Rowbottom]] cardinal, [[strongly Ramsey]] cardinal, [[virtually Ramsey]] cardinal
+
* [[Ramsey#Super Ramsey cardinal|super Ramsey]] cardinal
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$  
+
* [[Ramsey#Strongly Ramsey cardinal|strongly Ramsey]] cardinal
 +
* '''[[Ramsey]]''' cardinal, [[Jonsson | Jónsson]] cardinal, [[Rowbottom]] cardinal, [[Ramsey#Virtually Ramsey cardinal|virtually Ramsey]] cardinal
 +
* [[Ramsey#Almost Ramsey cardinal|almost Ramsey]] cardinal
 +
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$, [[Chang's conjecture]]
 +
* [[Ramsey#.24.5Calpha.24-iterable cardinal|$\omega_1$-iterable]] cardinal
 
* '''[[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]]
 
* [[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 $1\leq\alpha<\omega_1$
 
* [[remarkable]] cardinal
 
* [[remarkable]] cardinal
 +
* [[Ramsey#.24.5Calpha.24-iterable cardinal|weakly Ramsey]] cardinal
 
* [[ineffable]] cardinal, [[weakly ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy; [[completely ineffable]] cardinal
 
* [[ineffable]] cardinal, [[weakly ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy; [[completely ineffable]] cardinal
 
* [[subtle]] cardinal
 
* [[subtle]] cardinal
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* [[ORD is Mahlo|$\text{Ord}$ 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
 +
* [[Jäger's collapsing functions and ρ-inaccessible ordinals]]
 
* [[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
 
* [[inaccessible#Universes | Grothendieck universe axiom]], equivalent to the existence of a proper class of [[inaccessible]] cardinals
 
* [[inaccessible#Universes | Grothendieck universe axiom]], equivalent to the existence of a proper class of [[inaccessible]] cardinals

Revision as of 08:49, 7 February 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.