Kleene's O (original) (raw)
In set theory and computability theory, Kleene's is a canonical subset of the natural numbers when regarded as ordinal notations. It contains ordinal notations for every computable ordinal, that is, ordinals below Church–Kleene ordinal, . Since is the first ordinal not representable in a computable system of ordinal notations the elements of can be regarded as the canonical ordinal notations.
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| dbo:abstract | In set theory and computability theory, Kleene's is a canonical subset of the natural numbers when regarded as ordinal notations. It contains ordinal notations for every computable ordinal, that is, ordinals below Church–Kleene ordinal, . Since is the first ordinal not representable in a computable system of ordinal notations the elements of can be regarded as the canonical ordinal notations. Kleene (1938) described a system of notation for all computable ordinals (those less than the Church–Kleene ordinal). It uses a subset of the natural numbers instead of finite strings of symbols. Unfortunately, there is in general no effective way to tell whether some natural number represents an ordinal, or whether two numbers represent the same ordinal. However, one can effectively find notations which represent the ordinal sum, product, and power (see ordinal arithmetic) of any two given notations in Kleene's ; and given any notation for an ordinal, there is a computably enumerable set of notations which contains one element for each smaller ordinal and is effectively ordered. (en) |
| dbo:wikiPageExternalLink | https://www.ams.org/bull/1938-44-04/S0002-9904-1938-06720-1/ |
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| dbo:wikiPageRevisionID | 1123338044 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Computability_theory dbr:Computable_function dbr:Analytical_hierarchy dbr:Church–Kleene_ordinal dbr:Ordinal_arithmetic dbr:Ordinal_notation dbr:Stephen_Cole_Kleene dbr:Well-founded_relation dbr:Large_countable_ordinal dbc:Ordinal_numbers dbr:Natural_number dbr:Recursive_ordinal dbr:Recursively_enumerable_set dbr:Set_theory dbr:Partial_order dbr:Ordinal_notations |
| dbp:wikiPageUsesTemplate | dbt:Citation |
| dcterms:subject | dbc:Ordinal_numbers |
| rdf:type | yago:WikicatOrdinalNumbers yago:Abstraction100002137 yago:DefiniteQuantity113576101 yago:Measure100033615 yago:Number113582013 yago:OrdinalNumber113597280 |
| rdfs:comment | In set theory and computability theory, Kleene's is a canonical subset of the natural numbers when regarded as ordinal notations. It contains ordinal notations for every computable ordinal, that is, ordinals below Church–Kleene ordinal, . Since is the first ordinal not representable in a computable system of ordinal notations the elements of can be regarded as the canonical ordinal notations. (en) |
| rdfs:label | Kleene's O (en) |
| owl:sameAs | freebase:Kleene's O yago-res:Kleene's O wikidata:Kleene's O https://global.dbpedia.org/id/4pb74 |
| prov:wasDerivedFrom | wikipedia-en:Kleene's_O?oldid=1123338044&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Kleene's_O |
| is dbo:wikiPageWikiLink of | dbr:Kőnig's_lemma dbr:Ordinal_analysis dbr:Ordinal_notation dbr:Basis_theorem_(computability) dbr:Stephen_Cole_Kleene dbr:Computable_ordinal dbr:Large_countable_ordinal |
| is foaf:primaryTopic of | wikipedia-en:Kleene's_O |