Ramsey cardinal
 | This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages)
(Learn how and when to remove this template message)
|
In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by (Erdős Hajnal) and named after Frank P. Ramsey, whose theorem, called Ramsey's 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. This in turn implies the falsity of the Axiom of Constructibility of Kurt Gödel.
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.
Definition by κ-models
A regular cardinal κ is Ramsey if and only if[1] for any set A ⊂ κ, there is a transitive set M ⊨ ZFC− (i.e. ZFC without the axiom of powerset) of size κ with A ∈ M, and a nonprincipal ultrafilter U on the Boolean algebra P(κ) ∩ M such that:
- U is an M-ultrafilter: for any sequence ⟨Xβ : β < κ⟩ ∈ M of members of U, the diagonal intersection ΔXβ = {α < κ : ∀β < α(α ∈ Xβ)} ∈ U,
- U is weakly amenable: for any sequence ⟨Xβ : β < κ⟩ ∈ M of subsets of κ, the set {β < κ : Xβ ∈ U} ∈ M, and
- U is σ-complete: the intersection of any countable family of members of U is again in U.
References
- ↑ Gitman, Victoria (2008). "Ramsey-like cardinals". arXiv:0801.4723v2 [math.LO].
Bibliography
- Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.
- Erdős, Paul; Hajnal, András (1962), "Some remarks concerning our paper On the structure of set-mappings. Non-existence of a two-valued σ-measure for the first uncountable inaccessible cardinal", Acta Mathematica Academiae Scientiarum Hungaricae 13 (1–2): 223–226, doi:10.1007/BF02033641, ISSN 0001-5954
- Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3.
 |  |
