List of large cardinal properties

List of large cardinal properties

This page is a 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}, ... (see Aleph 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 (= Π11-indescribable) Πmn-indescribable , totally indescribable cardinals
* λ-unfoldable, unfoldable cardinals
* subtle cardinals
* almost ineffable, ineffable, "n"-ineffable, totally ineffable cardinals
* remarkable cardinals
* α-Erdős cardinals (for countable α), 0# (not a cardinal), γ-Erdős cardinals (for uncountable γ)
* almost Ramsey, Jónsson, Rowbottom, Ramsey, ineffably Ramsey cardinals
* measurable cardinals
* 0
* λ-strong, strong cardinals
* Woodin, weakly hyper-Woodin, Shelah, hyper-Woodin cardinals
* superstrong cardinals (=1-superstrong; for "n"-superstrong for "n"≥2 see further down.)
* subcompact, strongly compact (Woodin< strongly compact&le;supercompact), supercompact cardinals
* η-extendible, extendible cardinals
* Vopěnka cardinals
* "n"-superstrong ("n"&ge;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 cardinals (not consistent with the axiom of choice)
* 0=1 is (somewhat jokingly) listed as the ultimate large cardinal axiom by some authors.


*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 ( [ typescript] )
title=Higher Set Theory
DOI 10.1007/BFb0103096
ISBN =978-3-540-08926-1
DOI =10.1007/BFb0103104
pages= 99-275


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