In
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, the Erdős cardinal is the smallest cardinal such that
:
Existence of
zero sharp 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 ...
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 lea ...
, there is an -Erdős cardinal". In fact, for every
indiscernible satisfies "for every ordinal , there is an -Erdős cardinal in (the
Levy collapse to make countable)".
However, existence of an -Erdős cardinal implies existence of
zero sharp. If is the
satisfaction relation
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
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 constructi ...
, 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 imm ...
.
If κ is -Erdős, then it is -Erdős in every
transitive model satisfying " is countable".
See also
*
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 property. Existence of a cardinal number κ of a g ...
References
*
*
*
*
Large cardinals
Cardinal
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