# Category: Large cardinals

Wholeness axiom
In mathematics, the wholeness axiom is a strong axiom of set theory introduced by in 2000.
Equiconsistency
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 consi
Vopěnka's principle
In mathematics, 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, some members are similar to o
Indescribable cardinal
In mathematics, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to describe in some language Q. There are many different types of indescribable cardinals correspondi
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
Axiom of projective determinacy
In mathematical logic, projective determinacy is the special case of the axiom of determinacy applying only to projective sets. The axiom of projective determinacy, abbreviated PD, states that for any
Quasicompact cardinal
No description available.
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 on a cardinal κ, or more generally on any set. For
Zero sharp
In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. It is often encode
Solovay model
In the mathematical field of set theory, the Solovay model is a model constructed by Robert M. Solovay in which all of the axioms of Zermelo–Fraenkel set theory (ZF) hold, exclusive of the axiom of ch
Mahlo cardinal
In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by Paul Mahlo . As with all large cardinals, none of these varieties of Mahlo cardinal
Extender (set theory)
In set theory, an extender is a system of ultrafilters which represents an elementary embedding witnessing large cardinal properties. A nonprincipal ultrafilter is the most basic case of an extender.
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
Subcompact cardinal
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+) there is a non-trivial elementary embedding j:(H(
Zero dagger
In set theory, 0† (zero dagger) is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s. (The superscript † should be a dagger, but it appear
Rowbottom cardinal
In set theory, a Rowbottom cardinal, introduced by Rowbottom, is a certain kind of large cardinal number. An uncountable cardinal number is said to be - Rowbottom if for every function f: [κ]<ω → λ (w
Unfoldable cardinal
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 transitive model M of cardinality κ of ZFC-min
Worldly cardinal
In mathematical set theory, a worldly cardinal is a cardinal κ such that the rank Vκ is a model of Zermelo–Fraenkel set theory.
Tall cardinal
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 j : V → M with critical point κ such that
Woodin cardinal
In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number such that for all functions there exists a cardinal with and an elementary embedding from the Von Neumann universe into
Ramsey cardinal
In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by and named after Frank P. Ramsey, whose theorem establishes that ω enjoys a certain property that Ramsey cardi
Core model
In set theory, the core model is a definable inner model of the universe of all sets. Even though set theorists refer to "the core model", it is not a uniquely identified mathematical object. Rather,
List of large cardinal properties
This page includes a list of cardinals with large cardinal properties. It is arranged roughly in order of the consistency strength of the axiom asserting the existence of cardinals with the given prop
Normal measure
In set theory, a normal measure is a measure on a measurable cardinal κ such that the equivalence class of the identity function on κ maps to κ itself in the ultrapower construction. Equivalently, if
Berkeley cardinal
In set theory, Berkeley cardinals are certain large cardinals suggested by Hugh Woodin in a seminar at the University of California, Berkeley in about 1992. A Berkeley cardinal is a cardinal κ in a mo
Huge cardinal
In mathematics, a cardinal number κ is called huge if there exists an elementary embedding j : V → M from V into a transitive inner model M with critical point κ and Here, αM is the class of all seque
Grothendieck universe
In mathematics, a Grothendieck universe is a set U with the following properties: 1. * If x is an element of U and if y is an element of x, then y is also an element of U. (U is a transitive set.) 2.
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 ve
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 1. *
Shelah cardinal
In axiomatic set theory, Shelah cardinals are a kind of large cardinals. A cardinal is called Shelah iff for every , there exists a transitive class and an elementary embedding with critical point ; a
Ethereal cardinal
No description available.
Shrewd cardinal
In mathematics, a shrewd cardinal is a certain kind of large cardinal number introduced by, extending the definition of indescribable cardinals. For an ordinal λ, a cardinal number κ is called λ-shrew
Reinhardt cardinal
In set theory, a branch of mathematics, a Reinhardt cardinal is a kind of large cardinal. Reinhardt cardinals are considered under ZF (Zermelo–Fraenkel set theory without the Axiom of Choice), because
Reflecting cardinal
In set theory, a mathematical discipline, a reflecting cardinal is a cardinal number κ for which there is a normal ideal I on κ such that for every X∈I+, the set of α∈κ for which X reflects at α is in
Iterable cardinal
In mathematics, an iterable cardinal is a type of large cardinal introduced by Gitman, and Sharpe and Welch, and further studied by Gitman and Welch. Sharpe and Welch defined a cardinal κ to be iterab
Homogeneous (large cardinal property)
In set theory and in the context of a large cardinal property, a subset, S, of D is homogeneous for a function f if f is constant in finite subsets of S. More precisely, given a set D, let be the set
Ineffable cardinal
In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by . In the following definitions, will always be a regular uncountable cardinal
Kunen's inconsistency theorem
In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by Kenneth Kunen, shows that several plausible large cardinal axioms are inconsistent with the axiom of choice. Some conse
Inaccessible cardinal
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal κ is strongly inaccess
Strongly compact cardinal
In set theory, a branch of mathematics, a strongly compact cardinal is a certain kind of large cardinal. A cardinal κ is strongly compact if and only if every κ-complete filter can be extended to a κ-
Axiom of determinacy
In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological game
Hereditarily countable set
In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets. This inductive definition is well-founded and can be expressed in the language of first-o
Critical point (set theory)
In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not mapped to itself. Suppose that is an elementary em
Weakly compact cardinal
In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the stan
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 ⊆ M. Similarly,
Supercompact cardinal
In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties.
Strong cardinal
In set theory, a strong cardinal is a type of large cardinal. It is a weakening of the notion of a supercompact cardinal.
The Higher Infinite
The Higher Infinite: Large Cardinals in Set Theory from their Beginnings is a monograph in set theory by Akihiro Kanamori, concerning the history and theory of large cardinals, infinite sets character
Extendible cardinal
In mathematics, extendible cardinals are large cardinals introduced by , who was partly motivated by reflection principles. Intuitively, such a cardinal represents a point beyond which initial pieces
Laver function
In set theory, a Laver function (or Laver diamond, named after its inventor, Richard Laver) is a function connected with supercompact cardinals.
Moschovakis coding lemma
The Moschovakis coding lemma is a lemma from descriptive set theory involving sets of real numbers under the axiom of determinacy (the principle — incompatible with choice — that every two-player inte
Indestructibility
No description available.
Jónsson cardinal
In 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 if for every function f: [κ]<ω → κ the
Erdős cardinal
In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by Paul Erdős and András Hajnal. The Erdős cardinal κ(α) is defined to be the