Difference between revisions of "Lower attic"

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*  [[Aleph#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)-->
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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
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* [[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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** [[omega one chess| $\omega_1^{\mathfrak{Ch}}$]] = the supremum of the game values for white of the finite positions in infinite chess
 
** [[omega one chess| $\omega_1^{\mathfrak{Ch}}$]] = the supremum of the game values for white of the finite positions in infinite chess
 
* the [[Buchholz's ψ functions#Takeuti-Feferman-Buchholz ordinal|Takeuti-Feferman-Buchholz]] ordinal
 
* the [[Buchholz's ψ functions#Takeuti-Feferman-Buchholz ordinal|Takeuti-Feferman-Buchholz]] ordinal
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* the [[RHS0]] notation
 
* the [[Madore's ψ function#Bachmann-Howard ordinal|Bachmann-Howard]] ordinal
 
* the [[Madore's ψ function#Bachmann-Howard ordinal|Bachmann-Howard]] ordinal
 
* the [[Madore's ψ function#Large Veblen ordinal|large Veblen]] ordinal
 
* the [[Madore's ψ function#Large Veblen ordinal|large Veblen]] ordinal

Revision as of 11:46, 29 May 2022

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.