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| {{main|Large cardinal}}
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| This page includes a list of cardinals with [[large cardinal]] properties. It is arranged roughly in order of the consistency strength of the axiom asserting the existence of cardinals with the given property. Existence of a cardinal number κ of a given type implies the existence of cardinals of most of the types listed above that type, and for most listed cardinal descriptions φ of lesser consistency strength, [[Von Neumann universe|''V''<sub>κ</sub>]] satisfies "there is an unbounded class of cardinals satisfying φ".
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| The following table usually arranges cardinals in order of [[equiconsistency#Consistency strength|consistency strength]], with size of the cardinal used as a tiebreaker. In a few cases (such as strongly compact cardinals) the exact consistency strength is not known and the table uses the current best guess.
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| * "Small" cardinals: 0, 1, 2, ..., <math>\aleph_0, \aleph_1</math>,..., <math>\kappa = \aleph_{\kappa}</math>, ... (see [[Aleph number]])
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| * weakly and strongly [[inaccessible cardinal|inaccessible]], α-[[inaccessible cardinal|inaccessible]], and hyper inaccessible cardinals
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| * weakly and strongly [[Mahlo cardinal|Mahlo]], α-[[Mahlo cardinal|Mahlo]], and hyper Mahlo cardinals.
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| * [[reflecting cardinal|reflecting]] cardinals
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| * [[weakly compact cardinal|weakly compact]] (= Π{{su|p=1|b=1}}-indescribable), [[indescribable cardinal|Π{{su|p=''m''|b=''n''}}-indescribable]], [[indescribable cardinal|totally indescribable]] cardinals
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| * [[unfoldable cardinal|λ-unfoldable]], [[unfoldable cardinal|unfoldable]] cardinals, [[indescribable cardinal|ν-indescribable]] cardinals and [[shrewd cardinal|λ-shrewd]], [[shrewd cardinal|shrewd]] cardinals [not clear how these relate to each other].
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| * [[subtle cardinal|ethereal cardinals]], [[subtle cardinal]]s
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| * [[almost ineffable cardinal|almost ineffable]], [[ineffable cardinal|ineffable]], [[ineffable cardinal|''n''-ineffable]], [[ineffable cardinal|totally ineffable]] cardinals
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| * [[remarkable cardinal]]s
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| * [[Erdős cardinal|α-Erdős cardinal]]s (for [[countable]] α), [[zero sharp|0<sup>#</sup>]] (not a cardinal), [[Erdős cardinal|γ-Erdős cardinal]]s (for [[uncountable set|uncountable]] γ)
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| * [[almost Ramsey cardinal|almost Ramsey]], [[Jónsson cardinal|Jónsson]], [[Rowbottom cardinal|Rowbottom]], [[Ramsey cardinal|Ramsey]], [[ineffably Ramsey cardinal|ineffably Ramsey]] cardinals
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| * [[measurable cardinal]]s, [[zero dagger|0<sup>†</sup>]]
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| * [[strong cardinal|λ-strong]], [[strong cardinal|strong]] cardinals, [[tall cardinal|tall]] cardinals
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| * [[Woodin cardinal|Woodin]], [[weakly hyper-Woodin cardinal|weakly hyper-Woodin]], [[Shelah cardinal|Shelah]], [[hyper-Woodin cardinal|hyper-Woodin]] cardinals
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| * [[superstrong cardinal]]s (=1-superstrong; for ''n''-superstrong for ''n''≥2 see further down.)
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| * [[subcompact cardinal|subcompact]], [[strongly compact cardinal|strongly compact]] (Woodin< strongly compact≤supercompact), [[supercompact cardinal|supercompact]] cardinals
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| * [[extendible cardinal|η-extendible]], [[extendible cardinal|extendible]] cardinals
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| * [[Vopěnka cardinal]]s
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| * ''n''-[[superstrong cardinal|superstrong (''n''≥2)]], ''n''-[[huge cardinal|almost huge]], ''n''-[[huge cardinal|super almost huge]], ''n''-[[huge cardinal|huge]], ''n''-[[huge cardinal|superhuge]] cardinals (1-huge=huge, etc.)
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| * [[rank-into-rank]] (Axioms I3, I2, I1, and I0)
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| Finally, if there were a nontrivial [[elementary embedding]] from the entire [[von Neumann universe]] ''V'' into itself, ''j'':''V''→''V'', its [[Critical point (set theory)|critical point]] would be called a [[Reinhardt cardinal]]. It is provable in ZFC that there is no such embedding (and therefore no Reinhardt cardinals). However their existence has not yet been refuted in ZF alone (that is, without use of the [[axiom of choice]]). <!--
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| These do not really make sense, and it is not clear which should be first or last.
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| * [[contradiction|0=1]] is (somewhat jokingly) listed as the ultimate large cardinal axiom by some authors.
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| * The [[Consistency_strength#Consistency_strength|Strong]] [[Reflection principle|Reflection Principle]] of the [[Absolute Infinite]] [[Tav_(number)|Tav]] -->
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| ==References==
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| <references/> | |
| * {{cite book|author=Drake, F. R.|title=Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76)|publisher=Elsevier Science Ltd|year=1974|isbn=0-444-10535-2}}
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| * {{cite book|last=Kanamori|first=Akihiro|year=2003|authorlink=Akihiro Kanamori|publisher=Springer|title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings|edition=2nd|isbn=3-540-00384-3}}
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| *{{Cite book|last=Kanamori|first=Akihiro|first2=M. |last2=Magidor
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| |chapter=The evolution of large cardinal axioms in set theory
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| |series=Lecture Notes in Mathematics
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| |publisher =Springer Berlin / Heidelberg
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| |volume =669 ([http://math.bu.edu/people/aki/e.pdf typescript])
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| |title=Higher Set Theory
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| |year=1978
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| |isbn =978-3-540-08926-1
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| |doi =10.1007/BFb0103104
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| |pages=99–275|postscript=<!--None-->}}
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| * {{Cite journal|last=Solovay|first=Robert M.|first2=William N. |last2=Reinhardt|first3= Akihiro |last3=Kanamori|year=1978|title=Strong axioms of infinity and elementary embeddings|journal=Annals of Mathematical Logic|volume=13|issue=1|pages=73–116|authorlink=Robert M. Solovay|url=http://math.bu.edu/people/aki/d.pdf|doi=10.1016/0003-4843(78)90031-1|postscript=<!--None--> }}
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| ==External links==
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| *[http://cantorsattic.info/Upper_attic Cantor's attic]
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| [[Category:Large cardinals]]
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| [[Category:Mathematics-related lists|Large cardinals]]
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| [[cs:Velké kardinály]]
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