Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. High Quality Content by WIKIPEDIA articles! In set theory, an ordinal number is an admissible ordinal if L is an admissible set (that is, a transitive model of Kripke–Platek set theory); in other words, is admissible when is a limit ordinal and L 0-collection. The first two admissible ordinals are and omega_1^{mathrm{CK}} (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 which are constructed in a manner similar to the Church-Kleene ordinal but for Turing machines with oracles. One sometimes writes omega_alpha^ {mathrm{CK}} 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 which is highly parallel to that of (small) large cardinals (one can define recursively Mahlo cardinals, for example). But all these ordinals can still be countable.

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