Difference between revisions of "Lower attic"

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(ZFC-P<!--much above, much below...-->)
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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]]
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*  [[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)-->
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* [[Heights of models]] <!--(ZFC without powerset axiom) is much above $\Sigma_n$-admissible, much below ZFC (stable ordinals as part of ZFC have no consistency strength)-->
 
* The ordinals of [[infinite time Turing machines]], including   
 
* The ordinals of [[infinite time Turing machines]], including   
** [[infinite time Turing machines#Sigma | $\Sigma$]] = the supremum of the accidentally writable ordinals
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** [[infinite time Turing machines#Sigma | $\Sigma$]] = the supremum of the accidentally writable ordinals,
** [[infinite time Turing machines#zeta | $\zeta$]] = the supremum of the eventually writable ordinals  
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** [[infinite time Turing machines#zeta | $\zeta$]] = the supremum of the eventually writable ordinals,
** [[infinite time Turing machines#lambda | $\lambda$]] = the supremum of the writable ordinals,
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** [[infinite time Turing machines#lambda | $\lambda$]] = the supremum of the writable ordinals
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* [[bad]] ordinals
 +
* [[reflecting ordinal|reflecting]] ordinals
 
* [[admissible]] ordinals and [[Church-Kleene#relativized Church-Kleene ordinal | relativized Church-Kleene $\omega_1^x$]]
 
* [[admissible]] ordinals and [[Church-Kleene#relativized Church-Kleene ordinal | relativized Church-Kleene $\omega_1^x$]]
 
* [[Church-Kleene | Church-Kleene $\omega_1^{ck}$]], the supremum of the computable ordinals
 
* [[Church-Kleene | Church-Kleene $\omega_1^{ck}$]], the supremum of the computable ordinals
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* 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
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* [[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 17:22, 22 September 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.