Remarkable cardinal (original) (raw)
In mathematics, a remarkable cardinal is a certain kind of large cardinal number. A cardinal κ is called remarkable if for all regular cardinals θ > κ, there exist π, M, λ, σ, N and ρ such that 1. * π : M → Hθ is an elementary embedding 2. * M is countable and transitive 3. * π(λ) = κ 4. * σ : M → N is an elementary embedding with critical point λ 5. * N is countable and transitive 6. * ρ = M ∩ Ord is a regular cardinal in N 7. * σ(λ) > ρ 8. * M = HρN, i.e., M ∈ N and N ⊨ "M is the set of all sets that are hereditarily smaller than ρ"
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| dbo:abstract | In mathematics, a remarkable cardinal is a certain kind of large cardinal number. A cardinal κ is called remarkable if for all regular cardinals θ > κ, there exist π, M, λ, σ, N and ρ such that 1. * π : M → Hθ is an elementary embedding 2. * M is countable and transitive 3. * π(λ) = κ 4. * σ : M → N is an elementary embedding with critical point λ 5. * N is countable and transitive 6. * ρ = M ∩ Ord is a regular cardinal in N 7. * σ(λ) > ρ 8. * M = HρN, i.e., M ∈ N and N ⊨ "M is the set of all sets that are hereditarily smaller than ρ" Equivalently, is remarkable if and only if for every there is such that in some forcing extension , there is an elementary embedding satisfying . Although the definition is similar to one of the definitions of supercompact cardinals, the elementary embedding here only has to exist in , not in . (en) |
| dbo:wikiPageExternalLink | http://nylogic.org/wp-content/uploads/virtualLargeCardinals.pdf https://www.math.ucla.edu/~asl/bsl/0602/0602-003.ps |
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| gold:hypernym | dbr:Kind |
| rdf:type | yago:WikicatLargeCardinals yago:Bishop109857200 yago:Cardinal109894143 yago:CausalAgent100007347 yago:Clergyman109927451 yago:Leader109623038 yago:LivingThing100004258 yago:Object100002684 yago:Organism100004475 yago:Person100007846 yago:PhysicalEntity100001930 yago:Priest110470779 yago:YagoLegalActor yago:YagoLegalActorGeo yago:SpiritualLeader109505153 yago:Whole100003553 |
| rdfs:comment | In mathematics, a remarkable cardinal is a certain kind of large cardinal number. A cardinal κ is called remarkable if for all regular cardinals θ > κ, there exist π, M, λ, σ, N and ρ such that 1. * π : M → Hθ is an elementary embedding 2. * M is countable and transitive 3. * π(λ) = κ 4. * σ : M → N is an elementary embedding with critical point λ 5. * N is countable and transitive 6. * ρ = M ∩ Ord is a regular cardinal in N 7. * σ(λ) > ρ 8. * M = HρN, i.e., M ∈ N and N ⊨ "M is the set of all sets that are hereditarily smaller than ρ" (en) |
| rdfs:label | Remarkable cardinal (en) |
| owl:sameAs | freebase:Remarkable cardinal yago-res:Remarkable cardinal wikidata:Remarkable cardinal https://global.dbpedia.org/id/4tszW |
| prov:wasDerivedFrom | wikipedia-en:Remarkable_cardinal?oldid=1117467553&ns=0 |
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