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 conce ...
, an
ordinal number ''α'' is an admissible ordinal if
L''α'' is an
admissible set (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
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists a ...
and L
''α'' ⊧ Σ
0-collection.
[. See in particula]
p. 265
[.] The term was coined by Richard Platek in 1966.
The first two admissible ordinals are ω and
(the least
non-recursive ordinal, also called the
Church–Kleene ordinal
In mathematics, particularly set theory, non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using ordinal collapsing functions.
The Church–Kleene ordinal and varian ...
).
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.
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''. 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). 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. T ...
s.
Notice that ''α'' is an admissible ordinal if and only if ''α'' is a
limit ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists a ...
and there does not exist a ''γ'' < ''α'' for which there is a Σ
1(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
In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relev ...
*
Constructible universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It w ...
*
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. Infinite ...
References
Ordinal numbers
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