admissible ordinal
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In
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, an
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
''α'' is an admissible ordinal if L''α'' is an
admissible set In set theory, a discipline within mathematics, an admissible set is a transitive set A\, such that \langle A,\in \rangle is a model of Kripke–Platek set theory (Barwise 1975). The smallest example of an admissible set is the set of hereditaril ...
(that is, a
transitive model In mathematical set theory, a transitive model is a model of set theory that is standard and transitive. Standard means that the membership relation is the usual one, and transitive means that the model is a transitive set or class. Examples *An ...
of
Kripke–Platek set theory The Kripke–Platek set theory (KP), pronounced , is an axiomatic set theory developed by Saul Kripke and Richard Platek. The theory can be thought of as roughly the predicative part of ZFC and is considerably weaker than it. Axioms In its fo ...
); in other words, ''α'' is admissible when ''α'' is a limit ordinal and L''α'' ⊧ Σ0-collection.. See in particula
p. 265
. The term was coined by Richard Platek in 1966. The first two admissible ordinals are ω and \omega_1^ (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 In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
admissible ordinals are exactly those constructed in a manner similar to the Church–Kleene ordinal, but for Turing machines with
oracles An oracle is a person or agency considered to provide wise and insightful counsel or prophetic predictions, most notably including precognition of the future, inspired by deities. As such, it is a form of divination. Description The word ...
. One sometimes writes \omega_\alpha^ for the \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 In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least ...
(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 number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
s. 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(L''α'') mapping from ''γ'' onto ''α''.K. Devlin
An introduction to the fine structure of the constructible hierarchy
(1974) (p.38). Accessed 2021-05-06.
If ''M'' is a standard model of KP, then the set of ordinals in ''M'' is an admissible ordinal.


See also

* α-recursion theory * Large countable ordinals * Constructible universe *
Regular cardinal In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that \kappa is a regular cardinal if and only if every unbounded subset C \subseteq \kappa has cardinality \kappa. Infinit ...


References

Ordinal numbers {{settheory-stub