- List of large cardinal properties
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**list of some types of cardinals**; 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, V(κ) satisfies "there are unboundedly many cardinals satisfying φ".The following table usually arranges cardinals in order of 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.

* "Small" cardinals: 0, 1, 2, ..., $aleph\_0,\; aleph\_1$,..., $kappa\; =\; aleph\_\{kappa\}$, ... (seeAleph number )

* weakly and strongly inaccessible, α-inaccessible, and hyper inaccessible cardinals

* weakly and strongly Mahlo, α-Mahlo, and hyper Mahlo cardinals.

* reflecting cardinals

* weakly compact (= Π^{1}_{1}-indescribable) Π^{m}_{n}-indescribable , totally indescribable cardinals

* λ-unfoldable, unfoldable cardinals

*subtle cardinal s

* almost ineffable, ineffable, "n"-ineffable, totally ineffable cardinals

*remarkable cardinal s

* α-Erdős cardinals (forcountable α), 0^{#}(not a cardinal), γ-Erdős cardinals (for uncountable γ)

* almost Ramsey, Jónsson, Rowbottom, Ramsey, ineffably Ramsey cardinals

*measurable cardinal s

* 0^{†}

* λ-strong, strong cardinals

* Woodin, weakly hyper-Woodin, Shelah, hyper-Woodin cardinals

*superstrong cardinal s (=1-superstrong; for "n"-superstrong for "n"≥2 see further down.)

* subcompact, strongly compact (Woodin< strongly compact≤supercompact), supercompact cardinals

* η-extendible, extendible cardinals

*Vopěnka cardinal s

* "n"-superstrong ("n"≥2), "n"-almost huge, "n"-super almost huge, "n"-huge,"n"-superhuge cardinals (1-huge=huge, etc.)

*rank-into-rank (Axioms I3, I2, I1, and I0)

*Reinhardt cardinal s (not consistent with theaxiom of choice )

* 0=1 is (somewhat jokingly) listed as the ultimate large cardinal axiom by some authors.**References***

*

*citation|last=Kanamori|first=Akihiro|first2=M. |last2=Magidor

chapter=The evolution of large cardinal axioms in set theory

series= Lecture Notes in Mathematics

publisher =Springer Berlin / Heidelberg

ISSN = 1617-9692

volume =669 ( [*http://math.bu.edu/people/aki/e.pdf typescript*] )

title=Higher Set Theory

DOI 10.1007/BFb0103096

year=1978

ISBN =978-3-540-08926-1

DOI =10.1007/BFb0103104

pages= 99-275

*

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