Difference between revisions of "Lower attic"

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(the Takeuti-Feferman-Buchholz ordinal)
 
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Welcome to the lower attic, where we store the comparatively smaller notions of infinity. Roughly speaking, this is the realm of all brands of countable ordinals and their friends.  
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{{DISPLAYTITLE: The lower attic}}
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[[File:SagradaSpiralByDavidNikonvscanon.jpg | thumb | Sagrada Spiral photo by David Nikonvscanon]]
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[[Category:Lower attic]]
  
* <math>\Gamma</math>
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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.
* <math>\omega_1^{ck}</math>
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* <math>\epsilon_\alpha</math>
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* [[aleph one | $\omega_1$]], the first uncountable ordinal, and the other uncountable cardinals of the [[middle attic]]
* <math>\epsilon_1</math>
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* [[stable]] ordinals
* <math>\epsilon_0</math>
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* The ordinals of [[infinite time Turing machines]], including 
* Ordinals below <math>\omega^\omega</math>
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** [[infinite time Turing machines#Sigma | $\Sigma$]] = the supremum of the accidentally writable ordinals
* <math>\omega^3</math>
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** [[infinite time Turing machines#zeta | $\zeta$]] = the supremum of the eventually writable ordinals
* <math>\omega^2</math>
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** [[infinite time Turing machines#lambda | $\lambda$]] = the supremum of the writable ordinals,
* <math>\omega\cdot n+k</math>
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* [[admissible]] ordinals and [[Church-Kleene#relativized Church-Kleene ordinal | relativized Church-Kleene $\omega_1^x$]]
* <math>\omega\cdot2+1</math>
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* [[Church-Kleene | Church-Kleene $\omega_1^{ck}$]], the supremum of the computable ordinals
* <math>\omega\cdot2</math>
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* the [[omega one chess | omega one of chess]]
* <math>\omega+n</math>
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** [[omega one chess|$\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$]] = the supremum of the game values for white of all positions in infinite chess
* <math>\omega+2</math>
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** [[omega one chess| $\omega_1^{\mathfrak{Ch},c}$]] = the supremum of the game values for white of the computable positions in infinite chess
* <math>\omega+1</math>
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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
* <math>\omega</math>
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* the [[Buchholz's ψ functions#Takeuti-Feferman-Buchholz ordinal|Takeuti-Feferman-Buchholz]] ordinal
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* the [[Madore's ψ function#Bachmann-Howard ordinal|Bachmann-Howard]] ordinal
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* the [[Madore's ψ function#Large Veblen ordinal|large Veblen]] ordinal
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* the [[Madore's ψ function#Small Veblen ordinal|small Veblen]] ordinal
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* the [[Extended Veblen function]]
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* the [[Feferman-Schütte]] ordinal [[Feferman-Schütte | $\Gamma_0$]]
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* [[epsilon naught | $\epsilon_0$]] and the hierarchy of [[epsilon naught#epsilon_numbers | $\epsilon_\alpha$ numbers]]
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* [[indecomposable]] ordinal
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* 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 | $\epsilon_0$]]
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* [[Playroom#Hilbert's Grand Hotel | Hilbert's hotel]] and other toys in the [[playroom]]
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* [[omega | $\omega$]], the smallest infinity
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* down to the [[parlour]], where large finite numbers dream

Latest revision as of 13:37, 27 May 2018

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.