Difference between revisions of "Upper attic"

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(, completely remarkable)
(C(n))
Line 8: Line 8:
 
* [[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, [[L of V_lambda+1|I0 axiom]] and strengthenings
+
* '''[[rank into rank]]''' axioms, $C^{(n)}$-I3, $C^{(n)}$-I1, [[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, extendibility, supercompactness, strongness, etc...
+
* [[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|hyperhuge]], [[huge|ultrahuge]], [[superstrong|2-superstrong]] cardinal
 
* [[huge|almost huge]], '''[[huge]]''', [[huge|huge*]], [[huge|super almost huge]], [[huge|superhuge]], [[huge|hyperhuge]], [[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
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* [[Vopenka#Vopěnka cardinals|Vopěnka]] (=[[Woodin#Shelah cardinals|Woodin for supercompactness]]) cardinal
 
* [[Vopenka#Vopěnka cardinals|Vopěnka]] (=[[Woodin#Shelah cardinals|Woodin for supercompactness]]) cardinal
 
* [[Vopenka|Vopěnka's principle]]
 
* [[Vopenka|Vopěnka's principle]]
 +
* [[extendible|$C^{(n)}$-extendible]] hierarchy
 
* [[extendible]] cardinal, [[extendible|$\alpha$-extendible]] cardinal
 
* [[extendible]] cardinal, [[extendible|$\alpha$-extendible]] cardinal
 
* [[Woodin|Woodin for strong compactness]]
 
* [[Woodin|Woodin for strong compactness]]
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* 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]] cardinal
 
* [[Woodin#Shelah|Shelah]] cardinal
 
* [[Woodin#Shelah|Shelah]] cardinal

Revision as of 12:37, 14 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.