mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an Erdős cardinal, also called a partition cardinal is a certain kind 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 � ...

number introduced by . The Erdős cardinal is defined to be the least cardinal such that for every function there is a set of

order type In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y such ...

that is

homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...

for (if such a cardinal exists). In the notation of the

partition calculus In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom. R ...

, the Erdős cardinal is the smallest cardinal such that : Existence of

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 encoded as a subset of the integers (using Gödel ...

implies that the

constructible universe In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It w ...

satisfies "for every

countable ordinal In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least n ...

, there is an -Erdős cardinal". In fact, for every

indiscernible In mathematical logic, indiscernibles are objects that cannot be distinguished by any property or relation defined by a formula. Usually only first-order formulas are considered. Examples If ''a'', ''b'', and ''c'' are distinct and is a set ...

satisfies "for every ordinal , there is an -Erdős cardinal in (the

Levy collapse In mathematics, a collapsing algebra is a type of Boolean algebra sometimes used in forcing to reduce ("collapse") the size of cardinals. The posets used to generate collapsing algebras were introduced by Azriel Lévy in 1963. The collapsing alge ...

to make countable)". However, existence of an -Erdős cardinal implies existence of

. If is the satisfaction relation for (using ordinal parameters), then existence of zero sharp is equivalent to there being an -Erdős ordinal with respect to . And this in turn, the zero sharp implies the falsity of

axiom of constructibility The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as ''V'' = ''L'', where ''V'' and ''L'' denote the von Neumann universe and the construc ...

, of

Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imme ...

. If κ is -Erdős, then it is -Erdős in every

transitive model In mathematical set theory, a transitive model is a model of set theory that is standard and transitive. Standard means that the membership relation is the usual one, and transitive means that the model is a transitive set or class. Examples *An ...

satisfying " is countable".

References

* * * * Large cardinals

Cardinal Cardinal or The Cardinal may refer to: Animals * Cardinal (bird) or Cardinalidae, a family of North and South American birds **''Cardinalis'', genus of cardinal in the family Cardinalidae **''Cardinalis cardinalis'', or northern cardinal, the ...

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See also

References