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Set theory concept
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
Large_cardinal
This page includes a list of large cardinal properties in the mathematical field of set theory. It is arranged roughly in order of the consistency strength
List of large cardinal properties
List_of_large_cardinal_properties
Size of a set in mathematics
In mathematics, cardinality is an inherent property of sets, roughly meaning the number of individual objects they contain, which may be infinite. The
Cardinality
Set theory concept
measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal κ {\displaystyle
Measurable_cardinal
Type of infinite number in set theory
set theory, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal. A cardinal is a weakly inaccessible
Inaccessible_cardinal
Size of a possibly infinite set
In mathematics, a cardinal number, or cardinal for short, is a kind of number that measures the cardinality of a set, i.e., how many elements there are
Cardinal_number
Topics referred to by the same term
the College of Cardinals Cardinalidae, a family of North and South American birds Cardinal number in mathematics Large cardinal Cardinal direction, one
Cardinal
Branch of mathematics that studies sets
structure of the real number line to the study of the consistency of large cardinals. The basic notion of grouping objects has existed since at least the
Set_theory
Set-theoretic concept
In set theory, a Reinhardt cardinal is a kind of large cardinal. Reinhardt cardinals are considered under ZF (Zermelo–Fraenkel set theory without the
Reinhardt_cardinal
Extension of ideas in combinatorics to infinite sets
successors of singular cardinals. Write κ , λ {\displaystyle \kappa ,\lambda } for ordinals, m {\displaystyle m} for a cardinal number (finite or infinite)
Infinitary_combinatorics
Standard system of axiomatic set theory
attraction of large cardinal axioms is that they enable many results from ZF+AD to be established in ZFC adjoined by some large cardinal axiom. The Mizar
Zermelo–Fraenkel_set_theory
Type of large transfinite number
Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by Paul Mahlo (1911, 1912, 1913). As with all large cardinals
Mahlo_cardinal
Mathematical logic concept
universe, with important methodological consequences. The absoluteness of large cardinal axioms is also studied, with positive and negative results known. In
Absoluteness_(logic)
ω1-iterable cardinal, ω1-Erdos cardinal, Ramsey cardinal, and measurable cardinal. If zero sharp exists, then all stronger large cardinals (and indeed
Silver_cardinal
Basic framework of mathematics
heuristic reasons and that would decide the continuum hypothesis. Many large cardinal axioms were studied, but the hypothesis always remained independent
Foundations_of_mathematics
Set-theoretic concept
In set theory, Berkeley cardinals are certain large cardinals suggested by Hugh Woodin in a seminar at the University of California, Berkeley in about
Berkeley_cardinal
Kind of proposition in mathematics
axioms for set theory, such as some axioms asserting existence of large cardinals. In trying to formalize the argument for the reflection principle of
Reflection_principle
Topics referred to by the same term
Large cardinal, a property of certain transfinite numbers Large category, a category with a proper class of objects and morphisms (or both) Large diffeomorphism
Large
set theory, a Jónsson cardinal (named after Bjarni Jónsson) is a certain kind of large cardinal number. An uncountable cardinal number κ is said to be
Jónsson_cardinal
Large cardinal number that is hard to describe in a given language
set theory, a branch of mathematics, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to axiomatize in some language Q
Indescribable_cardinal
Type of large cardinal in set theory
weakly compact cardinal is a certain kind of cardinal number introduced by Erdős & Tarski (1961); weakly compact cardinals are large cardinals, meaning that
Weakly_compact_cardinal
Possible axiom for set theory in mathematics
the proposition that zero sharp exists and stronger large cardinal axioms (see list of large cardinal properties). Generalizations (i.e. weaker versions)
Axiom_of_constructibility
Large cardinal from set theory
"A new large cardinal and Laver sequences for extendibles", Fundamenta Mathematicae vol. 152 (1997). Asaf Karagila (2025). "Large Cardinals" (PDF). Morgenstern
Huge_cardinal
Axiom of set theory
proved in ZFC itself, but requires a mild large cardinal assumption (the existence of an inaccessible cardinal). The much stronger axiom of determinacy
Axiom_of_choice
Set whose elements all belong to another set
In this example, both sets are infinite, but the latter set has a larger cardinality (or power) than the former set. Another example in an Euler diagram:
Subset
Kind of large cardinal number
In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number λ {\displaystyle \lambda } such that for all functions f : λ → λ {\displaystyle
Woodin_cardinal
In set theory, a strongly compact cardinal is a certain kind of large cardinal. An uncountable cardinal κ is strongly compact if and only if every κ-complete
Strongly_compact_cardinal
Axiom of Zermelo-Fraenkel set theory
axiom of infinity is sometimes regarded as the first large cardinal axiom, and conversely large cardinal axioms are sometimes called[by whom?] stronger axioms
Axiom_of_infinity
Particular class of sets which can be described entirely in terms of simpler sets
Weakly Mahlo cardinals become strongly Mahlo. And more generally, any large cardinal property weaker than 0# (see the list of large cardinal properties)
Constructible_universe
Mathematical concept
In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by Erdős & Hajnal (1962) and named after Frank P. Ramsey, whose
Ramsey_cardinal
Subfield of mathematics
of large cardinals and determinacy. Large cardinals are cardinal numbers with particular properties so strong that the existence of such cardinals cannot
Mathematical_logic
Proposition in mathematical logic
the possible sizes of infinite sets. It states: There is no set whose cardinality is strictly between that of the integers and the real numbers. The name
Continuum_hypothesis
Large cardinal number
cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by Paul Erdős and András Hajnal (1958). A cardinal κ
Erdős_cardinal
Number of arguments required by a function
of n", though some are based on Latin cardinal numbers or ordinal numbers. For example, 1-ary is based on cardinal unus, rather than from distributive singulī
Arity
Set theory concept
sufficiently large cardinal number. In fact, by results of Moti Gitik, ZFC + ¬SCH is equiconsistent with ZFC + the existence of a measurable cardinal κ of Mitchell
Singular_cardinals_hypothesis
language large cardinal 1. A large cardinal is type of cardinal whose existence cannot be proved in ZFC. 2. A large large cardinal is a large cardinal that
Glossary_of_set_theory
Kind of large cardinal number
the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by Jensen & Kunen (1969). In the following
Ineffable_cardinal
Large cardinal from set theory
In set theory, a supercompact cardinal is a type of large cardinal independently introduced by Solovay and Reinhardt. They display a variety of reflection
Supercompact_cardinal
In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number. A cardinal κ {\displaystyle \kappa } is called
Subtle_cardinal
In set theory, a strong cardinal is a type of large cardinal. It is a weakening of the notion of a supercompact cardinal. If λ is any ordinal, κ is λ-strong
Strong_cardinal
Infinite cardinal number
\aleph _{0}} (read aleph-nought, aleph-zero, or aleph-null); the next larger cardinality of a well-ordered set is ℵ 1 , {\displaystyle \aleph _{1},} then ℵ
Aleph_number
Class of cardinal numbers
mathematics, limit cardinals are certain cardinal numbers. A cardinal number λ is a weak limit cardinal if λ is neither a successor cardinal nor zero. This
Limit_cardinal
Fundamental theorem in mathematical logic
However the situation is different when the language is of arbitrary large cardinality since then, though the completeness and compactness theorems remain
Gödel's_completeness_theorem
Collection of mathematical objects
the cardinality of finite sets is the basis of the concept of natural number, and predates for several thousands years the concept of sets. A large part
Set_(mathematics)
Process of repeating items in a self-similar way
Triptych, made in 1320. Its central panel contains the kneeling figure of Cardinal Stefaneschi, holding up the triptych itself as an offering. This practice
Recursion
Any one of the distinct objects that make up a set in set theory
known as cardinality; informally, this is the size of a set. In the above examples, the cardinality of the set A is 4, while the cardinality of set B
Element_of_a_set
In mathematics, an iterable cardinal is a type of large cardinal introduced by Gitman (2011), and Sharpe and Welch (2011), and further studied by Gitman
Iterable_cardinal
Large cardinal number
theory, a worldly cardinal is a cardinal κ such that the rank Vκ is a model of Zermelo–Fraenkel set theory. A strong limit cardinal κ is worldly if and
Worldly_cardinal
Ramsey cardinal Erdős cardinal Extendible cardinal Huge cardinal Hyper-Woodin cardinal Inaccessible cardinal Ineffable cardinal Mahlo cardinal Measurable
List of mathematical logic topics
List_of_mathematical_logic_topics
Vopěnka's principle is a large cardinal axiom. The intuition behind the axiom is that the set-theoretical universe is so large that in every proper class
Vopěnka's_principle
Ordered listing of items in collection
enumerations. Since set theorists work with infinite sets of arbitrarily large cardinalities, the default definition among this group of mathematicians of an
Enumeration
American mathematician (born 1948)
advanced to a study of Boolean relation theory, which attempts to justify large cardinal axioms by demonstrating their necessity for deriving certain propositions
Harvey Friedman (mathematician)
Harvey_Friedman_(mathematician)
Large cardinal property in set theory
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
Rank-into-rank
Being equally consistent
calibrated by large cardinals. For example: the negation of Kurepa's hypothesis is equiconsistent with the existence of an inaccessible cardinal, the non-existence
Equiconsistency
undefined term)p. 28 and is typically associated with a large cardinal notion. If Φ is a large cardinal notion, then the phrase "core model below Φ" refers
Core_model
Area of mathematical logic
infinite model for some infinite cardinal number, then it has a model of size κ for any sufficiently large cardinal number κ. Since two models of different
Model_theory
In mathematics, an unfoldable cardinal is a certain kind of large cardinal number. Formally, a cardinal number κ is λ-unfoldable if and only if for every
Unfoldable_cardinal
Theorem in descriptive set theory
theory, although they are relatively consistent with it, if certain large cardinals are consistent. A Gale–Stewart game is a two-player game of perfect
Borel_determinacy_theorem
Ordinals in mathematics and set theory
as large as the ones described in large cardinals, but they are large among those that have constructive notations (descriptions). Larger and larger ordinals
Large_countable_ordinal
Theorem in transfinite set theory
theorem, proved by Kenneth Kunen (1971), shows that several plausible large cardinal axioms are inconsistent with the axiom of choice. Some consequences
Kunen's_inconsistency_theorem
In mathematics, a tall cardinal is a large cardinal κ that is θ-tall for all ordinals θ, where a cardinal is called θ-tall if there is an elementary embedding
Tall_cardinal
consistency of a suitable large cardinal: Proper forcing axiom Open coloring axiom Martin's maximum Existence of 0# Singular cardinals hypothesis Projective
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Subset of the natural numbers in set theory
consistent). In other words, it is believed to be independent (see large cardinal for a discussion). It is usually formulated as follows: 0† exists if
0†
Concept in set theory
standard form of axiomatic set theory, but follows from a suitable large cardinal axiom. It was first introduced as a set of formulae in Silver's 1966
Zero_sharp
Statement that is taken to be true
as Morse–Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe is used, but in fact, most
Axiom
Subfield of set theory
relationship between determinacy and large cardinals. In general, stronger large cardinal axioms prove the determinacy of larger pointclasses, higher in the Wadge
Determinacy
Mathematical set formed from two given sets
ISBN 978-1-5275-8014-5. F. R. Drake, Set Theory: An Introduction to Large Cardinals, p. 24. Studies in Logic and the Foundations of Mathematics, vol. 76
Cartesian_product
Species of North American bird
The northern cardinal (Cardinalis cardinalis), also commonly known as the common cardinal, red cardinal, or simply cardinal, is a bird in the genus Cardinalis
Northern_cardinal
Formal system of logic
Löwenheim number of second-order logic is already larger than the first measurable cardinal, if such a cardinal exists. The Löwenheim number of first-order
Higher-order_logic
extendible cardinals are large cardinals introduced by Reinhardt (1974), who was partly motivated by reflection principles. Intuitively, such a cardinal represents
Extendible_cardinal
Set-theoretic concept
For each cardinal κ {\displaystyle \kappa } , there is a strongly inaccessible cardinal λ {\displaystyle \lambda } that is strictly larger than κ {\displaystyle
Grothendieck_universe
Theorem in mathematical logic
any theory that has arbitrarily large finite models, or a single infinite model, has models of arbitrary large cardinality (this is the Upward Löwenheim–Skolem
Compactness_theorem
inaccessible reflecting cardinal is not in general Mahlo however, see https://mathoverflow.net/q/212597. List of large cardinal properties Jech, Thomas
Reflecting_cardinal
Concept in model theory
Ehrenfeucht–Fraïssé games. Elementary embeddings are used in the study of large cardinals, including rank-into-rank. Two structures M and N of the same signature σ
Elementary_equivalence
In mathematics, a subcompact cardinal is a certain kind of large cardinal number. A cardinal number κ is subcompact if and only if for every A ⊂ H(κ+)
Subcompact_cardinal
One-to-one correspondence
bijection between them. More generally, two sets are said to have the same cardinal number if there exists a bijection between them. A bijective function from
Bijection
Type of cardinal number in mathematics
cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that κ {\displaystyle \kappa } is a regular cardinal if
Regular_cardinal
Smallest cardinal strictly greater in size than another cardinal
a successor operation on cardinal numbers in a similar way to the successor operation on the ordinal numbers. The cardinal successor coincides with the
Successor_cardinal
Collection of sets in mathematics that can be defined based on a property of its members
universal class), the class of all ordinal numbers, and the class of all cardinal numbers. One way to prove that a class is proper is to place it in bijection
Class_(set_theory)
Mathematical function such that every output has at least one input
left-total and right-total. The cardinality of the domain of a surjective function is greater than or equal to the cardinality of its codomain: If f : X →
Surjective_function
Topics referred to by the same term
to: Reinhardt University, Waleska, Georgia, USA Reinhardt cardinal, a kind of large cardinal Reinhardt (surname) Reinhardt Kristensen, Danish invertebrate
Reinhardt
Type of theory in mathematical logic
has cardinality κ and a theory is categorical in some uncountable cardinal greater than or equal to κ then it is categorical in all cardinalities greater
Categorical_theory
Set that is not a finite set
if and only if for every natural number, the set has a subset whose cardinality is that natural number. If the axiom of choice holds, then a set is infinite
Infinite_set
Mathematical set containing no elements
empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure
Empty_set
Set theory construction
173–210. doi:10.1002/malq.19570031302. ISSN 0942-5616. A. Kanamori, "Large Cardinals with Forcing". In Handbook of the History of Logic: Sets and Extensions
Solovay_model
Possible axiom for set theory
choices for ⟨b2, b4, b6, ...⟩ has the same cardinality as the continuum, which is larger than the cardinality of the proper initial portion { β ∈ J | β < α }
Axiom_of_determinacy
Set of elements common to all of some sets
\tau } ). Algebra of sets – Identities and relationships involving sets Cardinality – Size of a set in mathematics Complement – Set of the elements not in
Intersection_(set_theory)
American mathematician (born 1955)
to the theory of inner models and determinacy. A type of large cardinals, the Woodin cardinals, bears his name. In 2023, he was elected to the National
W._Hugh_Woodin
Mathematical concept for comparing objects
example of a theory which is ω-categorical, but not categorical for any larger cardinal number. An implication of model theory is that the properties defining
Equivalence_relation
Function that preserves distinctness
has at least as many elements as X , {\displaystyle X,} in the sense of cardinal numbers. In particular, if, in addition, there is an injection from Y
Injective_function
Term in mathematical logic
is consistent. The existence of strongly inaccessible cardinals The existence of large cardinals The non-existence of Kurepa trees The following statements
Independence (mathematical logic)
Independence_(mathematical_logic)
Mathematical set of all subsets of a set
infinite or not) always has strictly higher cardinality than the set itself (or informally, the power set must be larger than the original set). In particular
Power_set
Type of logical system
every model of φ; these models will in general be of arbitrarily large cardinality, and so logical consequence cannot be effectively verified by checking
First-order_logic
Set theory concept
explicitly after stage 5. The set Vω has the same cardinality as ω. The set Vω+1 has the same cardinality as the set of real numbers. In the standard Zermelo–Fraenkel
Von_Neumann_universe
Approximate distinct counting algorithm
cardinality of the distinct elements of a multiset requires an amount of memory proportional to the cardinality, which is impractical for very large data
HyperLogLog
Type of large cardinal number
theory, a Rowbottom cardinal, introduced by Rowbottom (1971), is a certain kind of large cardinal number. An uncountable cardinal number κ {\displaystyle
Rowbottom_cardinal
cardinal, but weaker than 2-iterable cardinal. Gitman, Victoria; Schindler, Ralf (2018-12-01). "Virtual large cardinals". Annals of Pure and Applied Logic
Virtually_extendible_cardinal
covering lemma is used to prove that the non-existence of certain large cardinals leads to the existence of a canonical inner model, called the core
Covering_lemma
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
Remarkable_cardinal
Theories in mathematical logic
example of a theory which is ω-categorical but not categorical for any larger cardinal. The equivalence relation ~ should not be confused with the identity
List_of_first-order_theories
LARGE CARDINAL
LARGE CARDINAL
Boy/Male
British, English
Large
Boy/Male
American, Australian, British, Celtic, Christian, English, French, German, Irish, Jamaican, Welsh
Prudent; Large Homestead; Large Settlement
Surname or Lastname
English (mainly Norfolk)
English (mainly Norfolk) : variant of Lark 1.
Girl/Female
British, English
Intelligent
Surname or Lastname
English
English : variant of Sark.German : unexplained.
Boy/Male
Australian, Welsh
Large Homestead; Large Settlement
Boy/Male
French
The red-haired one.
Girl/Female
American, Australian
Combination of Latonia and Ray
Surname or Lastname
English and French
English and French : metonymic occupational name for a boatman, from Middle English, Old French barge ‘boat’, ‘barge’.Dutch : variant of Berg.
Girl/Female
American, Australian, British, English
Skylark; Lark
Girl/Female
Swedish
From the sea.
Boy/Male
American, Australian, Chinese, Irish, Welsh
Large Homestead; Great Settlement; Large Village
Surname or Lastname
English and French
English and French : nickname (literal or ironic) meaning ‘generous’, from Middle English, Old French large ‘generous’, ‘free’ (Latin largus ‘abundant’). The English word came to acquire its modern sense only gradually during the Middle Ages; it is used to mean ‘ample in quantity’ in the 13th century, and the sense ‘broad’ first occurs in the 14th. This use is probably too late for the surname to have originated as a nickname for a fat man.
Boy/Male
Dutch
Large.
Boy/Male
Dutch Anglo Saxon
Tall.
Boy/Male
Tamil
Large quantity
Female
English
Short form of English Margaret, MARGE means "pearl."
Girl/Female
Persian American
Child of light. Famous Bearer: Margaret Thatcher, former Prime Minister of the United Kingdom.
Girl/Female
English
Lark.
Boy/Male
Hindu, Indian
Large
LARGE CARDINAL
LARGE CARDINAL
Girl/Female
Indian, Sanskrit
Morning
Boy/Male
Muslim
Father of faridoon a king
Boy/Male
Indian, Telugu
Study
Female
Egyptian
, the wife of King Aspalut.
Girl/Female
Muslim
Intelligent and beautiful
Boy/Male
Gujarati, Hindu, Indian
A Bird; Kuku
Boy/Male
Hindu
Motivation, Responsible
Girl/Female
Gujarati, Hindu, Indian, Kannada
Season
Boy/Male
Tamil
A part of Gauri parwati
Boy/Male
Australian, Dutch, French, German, Swedish
Brave Like a Bear
LARGE CARDINAL
LARGE CARDINAL
LARGE CARDINAL
LARGE CARDINAL
LARGE CARDINAL
n.
A larva.
n.
A movement or piece in largo time.
superl.
Prodigal in expending; lavish.
superl.
Having more than usual power or capacity; having broad sympathies and generous impulses; comprehensive; -- said of the mind and heart.
superl.
Free; unembarrassed.
n.
A large boat used by flag officers.
a.
Having large hands, Fig.: Taking, or giving, in large quantities; rapacious or bountiful.
a.
Made large or larger; extended; swollen.
n.
A large omnibus used for excursions.
n.
A shield or target.
adv.
Freely; licentiously.
superl.
Crossing the line of a ship's course in a favorable direction; -- said of the wind when it is abeam, or between the beam and the quarter.
superl.
Exceeding most other things of like kind in bulk, capacity, quantity, superficial dimensions, or number of constituent units; big; great; capacious; extensive; -- opposed to small; as, a large horse; a large house or room; a large lake or pool; a large jug or spoon; a large vineyard; a large army; a large city.
superl.
Abundant; ample; as, a large supply of provisions.
n.
A large, roomy boat for the conveyance of passengers or goods; as, a ship's barge; a charcoal barge.
n.
A musical note, formerly in use, equal to two longs, four breves, or eight semibreves.
n.
Border; margin; edge; verge.
superl.
Unrestrained by decorum; -- said of language.
a.
Having a large or generous heart or disposition; noble; liberal.
superl.
Full in statement; diffuse; full; profuse.