In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by Erdős & Hajnal (1962) and named after Frank P. Ramsey, whose theorem establishes that ω enjoys a certain property that Ramsey cardinals generalize to the uncountable case.
Let [κ]<ω denote the set of all finite subsets of κ. A cardinal number κ is called Ramsey if, for every function
f: [κ]<ω → {0, 1}
there is a set A of cardinality κ that is homogeneous for f. That is, for every n, the function f is constant on the subsets of cardinality n from A. A cardinal κ is called ineffably Ramsey if A can be chosen to be a stationary subset of κ. A cardinal κ is called virtually Ramsey if for every function
f: [κ]<ω → {0, 1}
there is C, a closed and unbounded subset of κ, so that for every λ in C of uncountable cofinality, there is an unbounded subset of λ that is homogenous for f; slightly weaker is the notion of almost Ramsey where homogenous sets for f are required of order type λ, for every λ < κ.
The existence of any of these kinds of Ramsey cardinal is sufficient to prove the existence of 0#, or indeed that 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 A ∉ I and for every function
f: [κ]<ω → {0, 1}
there is a set B ⊂ A not in I that is homogeneous for f. This is strictly stronger than κ being ineffably Ramsey.
The existence of a Ramsey cardinal implies the existence of 0# and this in turn implies the falsity of the Axiom of Constructibility of Kurt Gödel.
mathematics, a Ramseycardinal is a certain kind of large cardinal number introduced by Erdős & Hajnal (1962) and named after Frank P. Ramsey, whose theorem...
Silver and Solovay assumed the existence of a suitable large cardinal, such as a Ramseycardinal, and showed that with this extra assumption it is possible...
unfoldable cardinal is greater than the least indescribable cardinal.p.14 Assuming a Ramseycardinal exists, it is less than the least Ramseycardinal.p.3 A...
Arthur Michael Ramsey, Baron Ramsey of Canterbury, PC (14 November 1904 – 23 April 1988) was a British Church of England bishop and life peer. He served...
Vatican substitute for general affairs and later, a cardinal. Ramsey said that he tried to speak with Cardinal Edward Egan, then Archbishop of New York, about...
k {\displaystyle k} , "there is a k {\displaystyle k} -stationary Ramseycardinal", and S R P + {\displaystyle {\mathsf {SRP^{+}}}} represents the theory...
iterability is equivalent to ω1-iterability.) Gitman, Victoria (2011), "Ramsey-like cardinals I", Journal of Symbolic Logic, 76 (2): 519–540, arXiv:0801.4723...
Boniface Ramsey (born 6 October 1945) is an American Catholic priest who was ordained in 1973 as a member of the Dominican Order. From 1987 to 1996 Ramsey was...
Order-indiscernibles feature prominently in the theory of Ramseycardinals, Erdős cardinals, and zero sharp. Identity of indiscernibles Rough set Jech...
extends Ramsey's theorem to infinite cardinals ethereal cardinal An ethereal cardinal is a type of large cardinal similar in strength to subtle cardinals Euler...
Archbishop Gabriel Montalvo." Ramsey's 2000 letter was about complaints of sexual abuse of seminarians on the part of Cardinal Theodore McCarrick when he...
Florence, Massachusetts. He has three children. Laga graduated from Ramsey High School in Ramsey, New Jersey and attended Bergen Community College. the ESPN Baseball...
"Buster" Ramsey (March 16, 1920 – September 16, 2007) was an American football player for the College of William and Mary and Chicago Cardinals. He was...
school, Ramsey was rated as a four-star recruit, where he initially committed to play college football for the Stanford Cardinal. However, Ramsey decided...
Andre Alexander Ramsey (born July 24, 1987) is a former American football offensive tackle. He was signed by the Seattle Seahawks as a 7th round draft...
as his mentor Cardinal Désiré-Joseph Mercier, who had also sent him to Rome. Ordained to the priesthood on 4 September 1927 by Cardinal Jozef-Ernest van...
This paper, together with his thesis, "showed that Ramseycardinals were weaker than measurable cardinals, and that their existence implied the constructible...
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