Ramsey cardinal
Encyclopedia
In mathematics
, a Ramsey cardinal is a certain kind of large cardinal number introduced by and named after Frank P. Ramsey
.
With [κ]<ω denoting the set of all finite subsets of κ, a cardinal number
κ such that for every function
there is a set A of cardinality κ that is homogeneous for f (i.e.: for every n, f is constant on the subsets of cardinality n from A) is called Ramsey. A cardinal κ is called almost Ramsey if for every function
and for every λ < κ, there is a set of order type λ that is homogeneous for f.
The existence of a Ramsey cardinal is sufficient to prove the existence of 0#
. In fact, if κ is Ramsey, then every set with rank less than κ has a sharp.
Every measurable cardinal
is a Ramsey cardinal, and every Ramsey cardinal is a Rowbottom cardinal
.
A property intermediate in strength between Ramseyness and measurability
is existence of a κ-complete normal non-principal ideal
I on κ such that for every and for every function
there is a set B ⊂ A not in I that is homogeneous for f. If I is taken to be the ideal of nonstationary sets
, this property defines the ineffably Ramsey cardinals.
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...
, a Ramsey cardinal is a certain kind of large cardinal number introduced by and named after Frank P. Ramsey
Frank P. Ramsey
Frank Plumpton Ramsey was a British mathematician who, in addition to mathematics, made significant and precocious contributions in philosophy and economics before his death at the age of 26...
.
With [κ]<ω denoting the set of all finite subsets of κ, a cardinal number
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
κ such that for every function
- f: [κ]<ω → {0, 1}
there is a set A of cardinality κ that is homogeneous for f (i.e.: for every n, f is constant on the subsets of cardinality n from A) is called Ramsey. A cardinal κ is called almost Ramsey if for every function
- f: [κ]<ω → {0, 1}
and for every λ < κ, there is a set of order type λ that is homogeneous for f.
The existence of a Ramsey cardinal is sufficient to prove the existence of 0#
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...
. In fact, if κ is Ramsey, then every set with rank less than κ has a sharp.
Every measurable cardinal
Measurable cardinal
- Measurable :Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ...
is a Ramsey cardinal, and every Ramsey cardinal is a Rowbottom cardinal
Rowbottom cardinal
In set theory, a Rowbottom cardinal, introduced by , is a certain kind of large cardinal number.An uncountable cardinal number κ is said to be Rowbottom if for every function f: [κ]...
.
A property intermediate in strength between Ramseyness and measurability
Measurable cardinal
- Measurable :Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ...
is existence of a κ-complete normal non-principal ideal
Ideal (set theory)
In the mathematical field of set theory, an ideal is a collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal , and the union of any two elements of the ideal must also be in the ideal.More formally, given a set X, an...
I on κ such that for every and for every function
- f: [κ]<ω → {0, 1}
there is a set B ⊂ A not in I that is homogeneous for f. If I is taken to be the ideal of nonstationary sets
Stationary set
In mathematics, particularly in set theory and model theory, there are at least three notions of stationary set:-Classical notion:If \kappa \, is a cardinal of uncountable cofinality, S \subseteq \kappa \,, and S \, intersects every club set in \kappa \,, then S \, is called a stationary set....
, this property defines the ineffably Ramsey cardinals.