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- Strongly_compact_cardinal abstract "In mathematical set theory, a strongly compact cardinal is a certain kind of large cardinal number.A cardinal κ is strongly compact if and only if every κ-complete filter can be extended to a κ complete ultrafilter.Strongly compact cardinals were originally defined in terms of infinitary logic, where logical operators are allowed to take infinitely many operands. The logic on a regular cardinal κ is defined by requiring the number of operands for each operator to be less than κ; then κ is strongly compact if its logic satisfies an analog of the compactness property of finitary logic.Specifically, a statement which follows from some other collection of statements should also follow from some subcollection having cardinality less than κ.The property of strong compactness may be weakened by only requiring this compactness property to hold when the original collection of statements has cardinality below a certain cardinal λ; we may then refer to λ-compactness. A cardinal is weakly compact if and only if it is κ-compact; this was the original definition of that concept.Strong compactness implies measurability, and is implied by supercompactness. Given that the relevant cardinals exist, it is consistent with ZFC either that the first measurable cardinal is strongly compact, or that the first strongly compact cardinal is supercompact; these cannot both be true, however. A measurable limit of strongly compact cardinals is strongly compact, but the least such limit is not supercompact.The consistency strength of strong compactness is strictly above that of a Woodin cardinal. Some set theorists conjecture that existence of a strongly compact cardinal is equiconsistent with that of a supercompact cardinal. However, a proof is unlikely until a canonical inner model theory for supercompact cardinals is developed.Extendibility is a second-order analog of strong compactness.".
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- Strongly_compact_cardinal wikiPageOutDegree "14".
- Strongly_compact_cardinal wikiPageRevisionID "607170771".
- Strongly_compact_cardinal wikiPageWikiLink Cardinal_number.
- Strongly_compact_cardinal wikiPageWikiLink Category:Large_cardinals.
- Strongly_compact_cardinal wikiPageWikiLink Compactness_theorem.
- Strongly_compact_cardinal wikiPageWikiLink Extendible_cardinal.
- Strongly_compact_cardinal wikiPageWikiLink Infinitary_logic.
- Strongly_compact_cardinal wikiPageWikiLink Large_cardinal.
- Strongly_compact_cardinal wikiPageWikiLink Logical_connective.
- Strongly_compact_cardinal wikiPageWikiLink Logical_operator.
- Strongly_compact_cardinal wikiPageWikiLink Mathematics.
- Strongly_compact_cardinal wikiPageWikiLink Measurable_cardinal.
- Strongly_compact_cardinal wikiPageWikiLink Regular_cardinal.
- Strongly_compact_cardinal wikiPageWikiLink Set_theory.
- Strongly_compact_cardinal wikiPageWikiLink Supercompact_cardinal.
- Strongly_compact_cardinal wikiPageWikiLink Weakly_compact_cardinal.
- Strongly_compact_cardinal wikiPageWikiLink Woodin_cardinal.
- Strongly_compact_cardinal wikiPageWikiLinkText "Strongly compact cardinal".
- Strongly_compact_cardinal wikiPageWikiLinkText "strongly compact cardinal".
- Strongly_compact_cardinal wikiPageWikiLinkText "strongly compact".
- Strongly_compact_cardinal hasPhotoCollection Strongly_compact_cardinal.
- Strongly_compact_cardinal wikiPageUsesTemplate Template:Cite_book.
- Strongly_compact_cardinal wikiPageUsesTemplate Template:Settheory-stub.
- Strongly_compact_cardinal subject Category:Large_cardinals.
- Strongly_compact_cardinal hypernym Kind.
- Strongly_compact_cardinal comment "In mathematical set theory, a strongly compact cardinal is a certain kind of large cardinal number.A cardinal κ is strongly compact if and only if every κ-complete filter can be extended to a κ complete ultrafilter.Strongly compact cardinals were originally defined in terms of infinitary logic, where logical operators are allowed to take infinitely many operands.".
- Strongly_compact_cardinal label "Strongly compact cardinal".
- Strongly_compact_cardinal sameAs 강콤팩트_기수.
- Strongly_compact_cardinal sameAs m.068wgl.
- Strongly_compact_cardinal sameAs Q7624677.
- Strongly_compact_cardinal sameAs Q7624677.
- Strongly_compact_cardinal wasDerivedFrom Strongly_compact_cardinal?oldid=607170771.
- Strongly_compact_cardinal isPrimaryTopicOf Strongly_compact_cardinal.