Admissible Ordinal
978-613-1-82701-3
613182701X
72
2010-11-02
29.00 €
eng
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 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 α-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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