This page includes a list of
large cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least ...
properties in the mathematical field of
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
. 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 is an unbounded class of 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, ...,
,...,
, ... (see
Aleph number
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They were introduced by the mathematician Georg Cantor and are named after the symbol he used t ...
)
*
worldly cardinals
* 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-indescribable),
Π-indescribable,
totally indescribable cardinals,
ν-indescribable cardinals
*
λ-unfoldable,
unfoldable cardinals,
λ-shrewd,
shrewd cardinals (not clear how these relate to each other).
*
ethereal cardinals,
subtle cardinals
*
almost ineffable,
ineffable,
''n''-ineffable,
totally ineffable cardinals
*
remarkable cardinals
*
α-Erdős cardinals (for
countable
In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
α),
0# (not a cardinal),
γ-iterable,
γ-Erdős cardinals (for
uncountable
In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger tha ...
γ)
*
almost Ramsey,
Jónsson,
Rowbottom,
Ramsey,
ineffably Ramsey, completely Ramsey, strongly Ramsey, super Ramsey cardinals
*
measurable cardinal
In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure (mathematics), measure on a cardinal ''κ'', or more generally on any set. For a cardinal ''κ'', ...
s,
0†
*
λ-strong,
strong
Strong may refer to:
Education
* The Strong, an educational institution in Rochester, New York, United States
* Strong Hall (Lawrence, Kansas), an administrative hall of the University of Kansas
* Strong School, New Haven, Connecticut, United ...
cardinals,
tall 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≤supercompact),
supercompact,
hypercompact cardinals
*
η-extendible,
extendible cardinals
* almost high jump cardinals
*
Vopěnka cardinals, Shelah for supercompactness,
high jump cardinals, super high jump cardinals
* ''n''-
superstrong (''n''≥2), ''n''-
almost huge, ''n''-
super almost huge, ''n''-
huge, ''n''-
superhuge cardinals (1-huge=huge, etc.)
*
Wholeness axiom,
rank-into-rank
In set theory, a branch of mathematics, a rank-into-rank embedding is a large cardinal property defined by one of the following four axioms given in order of increasing consistency strength. (A set of rank < \lambda is one of the elements o ...
(Axioms I3, I2, I1, and I0)
The following even stronger large cardinal properties are not consistent with the axiom of choice, but their existence has not yet been refuted in ZF alone (that is, without use of the
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
).
*weakly Reinhardt cardinal,
Reinhardt cardinal,
Berkeley cardinal, super Reinhardt cardinal, totally Reinhardt cardinal
References
*
*
*
* {{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, doi-access=free
External links
Cantor's attic!-- old versio
Cantor's attic-->
some diagrams of large cardinal properties
*
Large cardinals
cs:Velké kardinály