Difference between revisions of "Upper attic"

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* The '''[[Kunen inconsistency]]''': [[Reinhardt]] cardinal, [[Kunen_inconsistency#Super_Reinhardt_cardinal | super Reinhardt]] cardinal, [[Berkeley]] cardinal
 
* The '''[[Kunen inconsistency]]''': [[Reinhardt]] cardinal, [[Kunen_inconsistency#Super_Reinhardt_cardinal | super Reinhardt]] cardinal, [[Berkeley]] cardinal
* '''[[Rank into rank]]''' cardinals $j:V_\lambda\to V_\lambda$, [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$, I0 cardinal [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]
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* '''[[Rank into rank]]''' axioms, [[L of V_lambda+1|I0 axiom]] and strengthenings
 
* The [[wholeness axioms]]
 
* The [[wholeness axioms]]
* [[n-fold variants | n-fold supercompact]], [[n-fold variants | n-fold strong]], [[n-fold variants | n-fold extendible]], [[n-fold variants | n-fold Woodin]]
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* [[n-fold variants]] of hugeness, extendibility, supercompactness, strongness, etc...
* '''[[huge|n-huge]]''' cardinal, [[huge|super n-huge]] cardinal, [[huge|ultra n-huge]] cardinal, [[superstrong|(n+1)-superstrong]] cardinal
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* '''[[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
 
* [[Vopenka | Vopěnka's principle]], [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin|Woodin for supercompactness]] cardinal

Revision as of 13:25, 8 December 2017

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.