In mathematics, specifically set theory, an ordinal <math>\alpha</math> is said to be recursive if there is a recursive binary relation <math>R</math> that well-orders a subset of the natural numbers and the order type of that ordering is <math>\alpha</math>. It is trivial to check that <math>\omega</math> is recursive, the successor of a recursive ordinal is recursive, and the set of all recursive ordinals is closed downwards.

PropertyValue
dbpprop:abstract
  • In mathematics, specifically set theory, an ordinal <math>\alpha</math> is said to be recursive if there is a recursive binary relation <math>R</math> that well-orders a subset of the natural numbers and the order type of that ordering is <math>\alpha</math>. It is trivial to check that <math>\omega</math> is recursive, the successor of a recursive ordinal is recursive, and the set of all recursive ordinals is closed downwards. We call the supremum of all recursive ordinals the Church-Kleene ordinal and denote it by <math>\omega^{CK}_1</math>. Since the recursive relations are parameterized by the natural numbers, the recursive ordinals are also parameterized by the natural numbers. Therefore, there are only countably many recursive ordinals. Thus, <math>\omega^{CK}_1</math> is countable. The recursive ordinals are exactly the ordinals that have an ordinal notation in Kleene's <math>\mathcal{O}</math>.
dbpprop:hasPhotoCollection
rdf:type
rdfs:comment
  • In mathematics, specifically set theory, an ordinal <math>\alpha</math> is said to be recursive if there is a recursive binary relation <math>R</math> that well-orders a subset of the natural numbers and the order type of that ordering is <math>\alpha</math>. It is trivial to check that <math>\omega</math> is recursive, the successor of a recursive ordinal is recursive, and the set of all recursive ordinals is closed downwards.
rdfs:label
  • Recursive ordinal
owl:sameAs
skos:subject
foaf:page
is dbpprop:redirect of
is owl:sameAs of