Difference between revisions of "Lower attic"
From Cantor's Attic
| Line 10: | Line 10: | ||
** [[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, | ||
| − | |||
* the [[Bachmann-Howard]] ordinal | * the [[Bachmann-Howard]] ordinal | ||
| − | * [[admissible]] ordinals | + | * [[admissible]] ordinals and [[admissible#relativized_admissible | relativized Church-Kleene $\omega_1^x$]] |
* Church-Kleene [[Church-Kleene omega_1 | $\omega_1^{ck}$]], the supremum of the computable ordinals | * Church-Kleene [[Church-Kleene omega_1 | $\omega_1^{ck}$]], the supremum of the computable ordinals | ||
* the [[Feferman-Schütte]] ordinal [[Feferman–Schütte | $\Gamma_0$]] | * the [[Feferman-Schütte]] ordinal [[Feferman–Schütte | $\Gamma_0$]] | ||
Revision as of 07:56, 30 December 2011
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
- The ordinals of infinite time Turing machines, including
- the Bachmann-Howard ordinal
- admissible ordinals and relativized Church-Kleene $\omega_1^x$
- Church-Kleene $\omega_1^{ck}$, the supremum of the computable ordinals
- the Feferman-Schütte ordinal $\Gamma_0$
- $\epsilon_0$ and the hierarchy of $\epsilon_\alpha$ numbers
- 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
- $\omega$, the smallest infinity
- down to the parlour, where large finite numbers dream
