Difference between revisions of "Upper attic"

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m ($\Sigma_n$-Mahlo)
(some new)
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* [[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 $\alpha$
 
* 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
 
* [[rank into rank|virtually rank-into-rank]] cardinal
 
* [[rank into rank|virtually rank-into-rank]] cardinal
 
* the [[Ramsey#.24.5Calpha.24-iterable cardinal|$n$-iterable]] and [[huge|virtually $n$-huge*]] hierarchy
 
* the [[Ramsey#.24.5Calpha.24-iterable cardinal|$n$-iterable]] and [[huge|virtually $n$-huge*]] hierarchy
 
* [[Woodin|virtually Shelah for supercompactness]] cardinal
 
* [[Woodin|virtually Shelah for supercompactness]] cardinal
 
* [[extendible|virtually extendible]], [[extendible|virtually $C^{(n)}$-extendible]] cardinal
 
* [[extendible|virtually extendible]], [[extendible|virtually $C^{(n)}$-extendible]] cardinal
* [[remarkable]] (=virtually supercompact), [[measurable|virtually measurable]], [[Ramsey|strategic $\omega$-Ramsey]] cardinals
+
* [[remarkable|$n$-remarkable]], [[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]]
 
* [[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]])

Revision as of 10:14, 11 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.