Difference between revisions of "Lower attic"

From Cantor's Attic
Jump to: navigation, search
(ZFC-P<!--much above, much below...-->)
m
Line 5: Line 5:
 
Welcome to the lower attic, where the countably infinite ordinals climb ever higher, one upon another, in an eternal self-similar reflecting ascent.
 
Welcome to the lower attic, where the countably infinite ordinals climb ever higher, one upon another, in an eternal self-similar reflecting ascent.
  
*  [[aleph one | $\omega_1$]], the first uncountable ordinal, and the other uncountable cardinals of the [[middle attic]]
+
*  [[Aleph#Aleph_one| $\omega_1$]], the first uncountable ordinal, and the other uncountable cardinals of the [[middle attic]]
 
* [[stable]] ordinals
 
* [[stable]] ordinals
 
* models of [[ZFC-P]] (ZFC without powerset axiom)<!--much above $\Sigma_n$-admissible, much below ZFC (stable ordinals as part of ZFC have no consistency strength)-->
 
* models of [[ZFC-P]] (ZFC without powerset axiom)<!--much above $\Sigma_n$-admissible, much below ZFC (stable ordinals as part of ZFC have no consistency strength)-->
Line 25: Line 25:
 
* the [[Feferman-Schütte]] ordinal [[Feferman-Schütte | $\Gamma_0$]]
 
* the [[Feferman-Schütte]] ordinal [[Feferman-Schütte | $\Gamma_0$]]
 
* [[epsilon naught | $\epsilon_0$]] and the hierarchy of [[epsilon naught#epsilon_numbers | $\epsilon_\alpha$ numbers]]
 
* [[epsilon naught | $\epsilon_0$]] and the hierarchy of [[epsilon naught#epsilon_numbers | $\epsilon_\alpha$ numbers]]
* [[indecomposable]] ordinal
+
* [[Limit_ordinal#Types_of_Limits|indecomposable]] ordinal
 
* the [[small countable ordinals]], such as [[small countable ordinals | $\omega,\omega+1,\ldots,\omega\cdot 2,\ldots,\omega^2,\ldots,\omega^\omega,\ldots,\omega^{\omega^\omega},\ldots$]] up to [[epsilon naught | $\epsilon_0$]]  
 
* the [[small countable ordinals]], such as [[small countable ordinals | $\omega,\omega+1,\ldots,\omega\cdot 2,\ldots,\omega^2,\ldots,\omega^\omega,\ldots,\omega^{\omega^\omega},\ldots$]] up to [[epsilon naught | $\epsilon_0$]]  
 
* [[Playroom#Hilbert's Grand Hotel | Hilbert's hotel]] and other toys in the [[playroom]]
 
* [[Playroom#Hilbert's Grand Hotel | Hilbert's hotel]] and other toys in the [[playroom]]
 
* [[omega | $\omega$]], the smallest infinity
 
* [[omega | $\omega$]], the smallest infinity
 
* down to the [[parlour]], where large finite numbers dream
 
* down to the [[parlour]], where large finite numbers dream

Revision as of 16:57, 30 July 2021

Sagrada Spiral photo by David Nikonvscanon

Welcome to the lower attic, where the countably infinite ordinals climb ever higher, one upon another, in an eternal self-similar reflecting ascent.