Metraipolis › archive › dc_0809475df001

Admissible ordinal — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Admissible_ordinal
License (as archived): CC-BY-SA-4.0
Archived: 2026-11-07T14:56:40.820Z
SHA-256 of the archived text: 05f5db3d7a23ce98f7177cc84d38ffa0b00c59540ee96ecd1c9d976c45f15c3f
Stable URL of this page: https://metraipolis.com/documento/dc_0809475df001

How to cite

Admissible ordinal — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Admissible_ordinal (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_0809475df001 (Metraipolis archive dc_0809475df001, sha256 05f5db3d7a23ce98f7177cc84d38ffa0b00c59540ee96ecd1c9d976c45f15c3f)

BibTeX

@misc{dc0809475df001,
  title = {Admissible ordinal — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Admissible_ordinal},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_0809475df001; sha256 05f5db3d7a23ce98f7177cc84d38ffa0b00c59540ee96ecd1c9d976c45f15c3f},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Admissible_ordinal. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In set theory, an ordinal number α is an admissible ordinal if Lα is an admissible set (that is, a transitive model of Kripke–Platek set theory); in other words, α is admissible when α is a limit ordinal and Lα ⊧ Σ0-collection. The term was coined by Richard Platek in 1966. The first two admissible ordinals are ω and ω 1 C K {\displaystyle \omega _{1}^{\mathrm {CK} }} (the least nonrecursive ordinal, also called the Church–Kleene ordinal). Any regular uncountable cardinal is an admissible ordinal. By a theorem of Sacks, the countable admissible ordinals are exactly those constructed in a manner similar to the Church–Kleene ordinal, but for Turing machines with oracles. One sometimes writes ω α C K {\displaystyle \omega _{\alpha }^{\mathrm {CK} }} for the α {\displaystyle \alpha } -th ordinal that is either admissible or a limit of admissibles; an ordinal that is both is called recursively inaccessible. There exists a theory of large ordinals in this manner that is highly parallel to that of (small) large cardinals (one can define recursively Mahlo ordinals, for example). But all these ordinals are still countable. Therefore, admissible ordinals seem to be the recursive analogue of regular cardinal numbers. Notice that α is an admissible ordinal if and only if α is a limit ordinal and there does not exist a γ < α for which there is a Σ1(Lα) mapping from γ onto α. α {\displaystyle \alpha } is an admissible ordinal if and only if there is a standard model M {\displaystyle M} of KP whose set of ordinals is α {\displaystyle \alpha } , in fact this may be take as the definition of admissibility. The α {\displaystyle \alpha } th admissible ordinal is sometimes denoted by τ α {\displaystyle \tau _{\alpha }} p. 174 or τ α 0 {\displaystyle \tau _{\alpha }^{0}} . The Friedman–Jensen–Sacks theorem states that a countable α {\displaystyle \alpha } is admissible if and only if there exists some A ⊆ ω {\displaystyle A\subseteq \omega } such that α {\displaystyle \alpha } is the least ordinal not recursive in A {\displaystyle A} . Equivalently, for any countable admissible α {\displaystyle \alpha } , there is an A ⊆ N {\displaystyle A\subseteq \mathbb {N} } making α {\displaystyle \alpha } minimal such that ⟨ L α , ∈ , A ⟩ {\displaystyle \langle L_{\alpha },\in ,A\rangle } is an admissible structure.p. 264

Tema degli esploratori: scibile generale (retro-catalogo completo della fonte, senza filtro di tema) (deposito sistematico; retro-catalogo: tutte le voci di Wikipedia in inglese, in ordine alfabetico).
Fonte: Wikipedia in inglese, voce «Admissible ordinal» — https://en.wikipedia.org/wiki/Admissible_ordinal
Wikidata Q4683804: «ordinal number α such that L_α is an admissible set (transitive model of Kripke–Platek set theory)» (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q4683804
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Admissible_ordinal?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1347479847, consultata il 2026-11-07T14:51:52.834Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 937b9a175ae53f0a044c4096b005a550af990026b14ad1259121ceadd1b72a5e