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. [1] [2] The term was coined by Richard Platek in 1966. [3]
The first two admissible ordinals are ω and (the least nonrecursive ordinal, also called the Church–Kleene ordinal). [2] 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. [1] One sometimes writes for the -th ordinal that is either admissible or a limit of admissibles; an ordinal that is both is called recursively inaccessible. [4] 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). [5] 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. [6] is an admissible ordinal iff there is a standard model of KP whose set of ordinals is , in fact this may be take as the definition of admissibility. [7] [8] The th admissible ordinal is sometimes denoted by [9] [8]p. 174 or . [10]
The Friedman-Jensen-Sacks theorem states that countable is admissible iff there exists some such that is the least ordinal not recursive in . [11]
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. [1] [2] The term was coined by Richard Platek in 1966. [3]
The first two admissible ordinals are ω and (the least nonrecursive ordinal, also called the Church–Kleene ordinal). [2] 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. [1] One sometimes writes for the -th ordinal that is either admissible or a limit of admissibles; an ordinal that is both is called recursively inaccessible. [4] 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). [5] 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. [6] is an admissible ordinal iff there is a standard model of KP whose set of ordinals is , in fact this may be take as the definition of admissibility. [7] [8] The th admissible ordinal is sometimes denoted by [9] [8]p. 174 or . [10]
The Friedman-Jensen-Sacks theorem states that countable is admissible iff there exists some such that is the least ordinal not recursive in . [11]