Difference between revisions of "Lower attic"
From Cantor's Attic
| (5 intermediate revisions by the same user not shown) | |||
| Line 9: | Line 9: | ||
* [[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)--> | * [[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 | + | ** [[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#zeta | $\zeta$]] = the supremum of the eventually writable ordinals, |
| − | ** [[infinite time Turing machines#lambda | $\lambda$]] = the supremum of the 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 $\omega_1^x$]] | * [[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 | ||
| Line 18: | Line 20: | ||
** [[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| $\omega_1^{\mathfrak{Ch},c}$]] = 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 | ** [[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 | + | * [[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#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 | ||
Latest revision as of 20:29, 1 July 2022
Welcome to the lower attic, where the countably infinite ordinals climb ever higher, one upon another, in an eternal self-similar reflecting ascent.
- $\omega_1$, the first uncountable ordinal, and the other uncountable cardinals of the middle attic
- stable ordinals
- Heights of models
- The ordinals of infinite time Turing machines, including
- bad ordinals
- reflecting ordinals
- admissible ordinals and relativized Church-Kleene $\omega_1^x$
- Church-Kleene $\omega_1^{ck}$, the supremum of the computable ordinals
- the omega one of chess
- $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$ = the supremum of the game values for white of all positions in infinite chess
- $\omega_1^{\mathfrak{Ch},c}$ = the supremum of the game values for white of the computable positions in infinite 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 Takeuti-Feferman-Buchholz ordinal
- the RHS0 notation
- the Bachmann-Howard ordinal
- the large Veblen ordinal
- the small Veblen ordinal
- the Extended Veblen function
- the Feferman-Schütte ordinal $\Gamma_0$
- $\epsilon_0$ and the hierarchy of $\epsilon_\alpha$ numbers
- indecomposable ordinal
- the small countable ordinals, such as $\omega,\omega+1,\ldots,\omega\cdot 2,\ldots,\omega^2,\ldots,\omega^\omega,\ldots,\omega^{\omega^\omega},\ldots$ up to $\epsilon_0$
- Hilbert's hotel and other toys in the playroom
- $\omega$, the smallest infinity
- down to the parlour, where large finite numbers dream
