Difference between revisions of "Upper attic"

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m (remarkable (=virtually supercompact))
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* $\alpha$-[[filter property]], [[Ramsey|$\alpha$-Ramsey]] cardinal (for $\omega < \alpha < \kappa$), [[Ramsey|almost fully Ramsey]] (=[[Ramsey|$<κ$-Ramsey]]) cardinal  
 
* $\alpha$-[[filter property]], [[Ramsey|$\alpha$-Ramsey]] cardinal (for $\omega < \alpha < \kappa$), [[Ramsey|almost fully Ramsey]] (=[[Ramsey|$<κ$-Ramsey]]) cardinal  
 
* [[Ramsey|$\Pi_\alpha$-Romsey]], [[Ramsey|completely Romsey]] (=[[Ramsey|$ω$-very Ramsey]]), [[Ramsey|$\alpha$-hyper completely Romsey]], [[Ramsey|super completely Romsey]] cardinals
 
* [[Ramsey|$\Pi_\alpha$-Romsey]], [[Ramsey|completely Romsey]] (=[[Ramsey|$ω$-very Ramsey]]), [[Ramsey|$\alpha$-hyper completely Romsey]], [[Ramsey|super completely Romsey]] cardinals
* [[Ramsey|Mahlo–Ramsey]] cardinal
+
* [[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]]''' cardinal, [[Jonsson|Jónsson]] cardinal, [[Rowbottom]] cardinal, [[Ramsey#Virtually Ramsey cardinal|virtually Ramsey]] cardinal
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* [[Erdos|$\omega_1$-Erdős]] cardinal and [[Erdos|$\gamma$-Erdős]] cardinals for uncountable $\gamma$, [[Chang's conjecture]]
 
* [[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, [[Ramsey|$(\omega, \omega_1)$-Ramsey]] cardinal
 
* [[Ramsey#.24.5Calpha.24-iterable cardinal|$\omega_1$-iterable]] cardinal, [[Ramsey|$(\omega, \omega_1)$-Ramsey]] 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]]
 
* [[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 $\alpha$

Revision as of 21:51, 6 June 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.