Erdos cardinal
Encyclopedia
In mathematics
, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by .
The Erdős cardinal κ(α) is defined to be the least cardinal such that for every function } there is a set of order type α that is homogeneous for f (if such a cardinal exists). In the notation of the partition calculus, the Erdős cardinal κ(α) is the smallest cardinal such that
Existence of zero sharp
implies that the constructible universe L satisfies "for every countable ordinal α, there is an α-Erdős cardinal". In fact, for every indiscernible κ, Lκ satisfies "for every ordinal α, there is an α-Erdős cardinal in Coll(ω, α) (the Levy collapse to make α countable)".
However, existence of an ω1-Erdős cardinal implies existence of zero sharp. If f is the satisfaction relation for L (using ordinal parameters), then existence of zero sharp is equivalent to there being an ω1-Erdős ordinal with respect to f.
If κ is α-Erdős, then it is α-Erdős in every transitive model satisfying "α is countable".
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by .
The Erdős cardinal κ(α) is defined to be the least cardinal such that for every function } there is a set of order type α that is homogeneous for f (if such a cardinal exists). In the notation of the partition calculus, the Erdős cardinal κ(α) is the smallest cardinal such that
- κ(α) → (α)<ω
Existence of zero sharp
Zero sharp
In the mathematical discipline of set theory, 0# is the set of true formulas about indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers , or as a subset of the hereditarily finite sets, or as a real number...
implies that the constructible universe L satisfies "for every countable ordinal α, there is an α-Erdős cardinal". In fact, for every indiscernible κ, Lκ satisfies "for every ordinal α, there is an α-Erdős cardinal in Coll(ω, α) (the Levy collapse to make α countable)".
However, existence of an ω1-Erdős cardinal implies existence of zero sharp. If f is the satisfaction relation for L (using ordinal parameters), then existence of zero sharp is equivalent to there being an ω1-Erdős ordinal with respect to f.
If κ is α-Erdős, then it is α-Erdős in every transitive model satisfying "α is countable".