Difference between revisions of "Lower attic"

From Cantor's Attic
Jump to: navigation, search
(Created page with "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 friend...")
 
 
(82 intermediate revisions by 10 users not shown)
Line 1: Line 1:
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.  
+
{{DISPLAYTITLE: The lower attic}}
 +
[[File:SagradaSpiralByDavidNikonvscanon.jpg | thumb | Sagrada Spiral photo by David Nikonvscanon]]
 +
[[Category:Lower attic]]
  
*  
+
Welcome to the lower attic, where the countably infinite ordinals climb ever higher, one upon another, in an eternal self-similar reflecting ascent.
*
+
 
*
+
* [[Aleph#Aleph_one| $\omega_1$]], the first uncountable ordinal, and the other uncountable cardinals of the [[middle attic]]
*
+
* [[stable]] ordinals
*  
+
* [[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 
*
+
** [[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,
*  
+
** [[infinite time Turing machines#lambda | $\lambda$]] = the supremum of the writable ordinals
*
+
* [[bad]] ordinals
*
+
* [[reflecting ordinal|reflecting]] ordinals
*
+
* [[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
*
+
* the [[omega one chess | omega one of chess]]
* <math>Insert formula here</math>
+
** [[omega one chess|$\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$]] = the supremum of the game values for white of all positions in infinite chess
* <math>\omega</math>
+
** [[omega one chess| $\omega_1^{\mathfrak{Ch},c}$]] = the supremum of the game values for white of the computable 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
 +
* [[Buchholz's ψ functions]], the [[Buchholz's ψ functions#Takeuti-Feferman-Buchholz ordinal|Takeuti-Feferman-Buchholz]] ordinal
 +
* the [[RHS0]] notation
 +
* 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#Small Veblen ordinal|small Veblen]] ordinal
 +
* the [[Extended Veblen function]]
 +
* the [[Feferman-Schütte]] ordinal [[Feferman-Schütte | $\Gamma_0$]]
 +
* [[epsilon naught | $\epsilon_0$]] and the hierarchy of [[epsilon naught#epsilon_numbers | $\epsilon_\alpha$ numbers]]
 +
* [[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 | $\epsilon_0$]]
 +
* [[Playroom#Hilbert's Grand Hotel | Hilbert's hotel]] and other toys in the [[playroom]]
 +
* [[omega | $\omega$]], the smallest infinity
 +
* down to the [[parlour]], where large finite numbers dream

Latest revision as of 20:29, 1 July 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.