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- Admissible_ordinal abstract "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 first two admissible ordinals are ω and (the least non-recursive 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 for the -th ordinal which is either admissible or a limit of admissibles; an ordinal which 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 cardinals, 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. If M is a standard model of KP, then the set of ordinals in M is an admissible ordinal.".
- Admissible_ordinal wikiPageID "4635859".
- Admissible_ordinal wikiPageRevisionID "414984799".
- Admissible_ordinal hasPhotoCollection Admissible_ordinal.
- Admissible_ordinal subject Category:Ordinal_numbers.
- Admissible_ordinal type Abstraction100002137.
- Admissible_ordinal type DefiniteQuantity113576101.
- Admissible_ordinal type Measure100033615.
- Admissible_ordinal type Number113582013.
- Admissible_ordinal type OrdinalNumber113597280.
- Admissible_ordinal type OrdinalNumbers.
- Admissible_ordinal comment "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 first two admissible ordinals are ω and (the least non-recursive ordinal, also called the Church–Kleene ordinal).".
- Admissible_ordinal label "Admissible ordinal".
- Admissible_ordinal sameAs m.0cdstr.
- Admissible_ordinal sameAs Q4683804.
- Admissible_ordinal sameAs Q4683804.
- Admissible_ordinal sameAs Admissible_ordinal.
- Admissible_ordinal wasDerivedFrom Admissible_ordinal?oldid=414984799.
- Admissible_ordinal isPrimaryTopicOf Admissible_ordinal.