This page includes a list of cardinals with
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. 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
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
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
In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other".
In general, it is not ...
, 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 that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named ...
)
*
worldly cardinal In mathematical set theory, a worldly cardinal is a cardinal κ such that the rank ''V''κ is a model of Zermelo–Fraenkel set theory.
Relationship to inaccessible cardinals
By Zermelo's theorem on inaccessible cardinals, every inaccessible c ...
s
* 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
*
λ-unfoldable,
unfoldable cardinals,
ν-indescribable cardinals and
λ-shrewd,
shrewd cardinals (not clear how these relate to each other).
*
ethereal cardinals,
subtle cardinal In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number.
A cardinal ''κ'' is called subtle if for every closed and unbounded ''C'' ⊂ ''κ'' and for every sequence ''A'' of length ''κ' ...
s
*
almost ineffable,
ineffable,
''n''-ineffable,
totally ineffable cardinals
*
remarkable cardinal In mathematics, a remarkable cardinal is a certain kind of large cardinal number.
A cardinal ''κ'' is called remarkable if for all regular cardinals ''θ'' > ''κ'', there exist ''π'', ''M'', ''λ'', ''σ'', ''N'' and ''ρ'' such that
# ''π'' : ...
s
*
α-Erdős cardinals (for
countable
In mathematics, a set is countable if either it is 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 from it into the natural number ...
α),
0# (not a cardinal),
γ-iterable,
γ-Erdős cardinals (for
uncountable γ)
*
almost Ramsey,
Jónsson Jónsson is a surname of Icelandic origin, meaning ''son of Jón''. In Icelandic names, the name is not strictly a surname, but a patronymic. The name refers to:
*Arnar Jónsson (actor) (born 1943), Icelandic actor
*Arnar Jónsson (basketball) (bor ...
,
Rowbottom,
Ramsey,
ineffably Ramsey, completely Ramsey, strongly Ramsey, super Ramsey cardinals
*
measurable cardinals,
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 Sta ...
cardinals,
tall cardinals
*
Woodin,
weakly hyper-Woodin,
Shelah,
hyper-Woodin cardinals
*
superstrong cardinal In mathematics, a cardinal number κ is called superstrong if and only if there exists an elementary embedding ''j'' : ''V'' → ''M'' from ''V'' into a transitive inner model ''M'' with critical point κ and V_ ⊆ ''M''.
Similar ...
s (=1-superstrong; for ''n''-superstrong for ''n''≥2 see further down.)
*
subcompact
Subcompact car is a North American classification for cars smaller than a compact car. It is broadly equivalent to the B-segment (Europe), supermini (Great Britain) or A0-class (China) classifications.
According to the U.S. Environmental Pro ...
,
strongly compact (Woodin< strongly compact≤supercompact),
supercompact,
hypercompact cardinals
*
η-extendible,
extendible cardinals
*
Vopěnka cardinals, Shelah for supercompactness,
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 (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, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
).
*
Reinhardt cardinal,
Berkeley 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