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Every analytic set is Ramsey

Published online by Cambridge University Press:  12 March 2014

Jack Silver
Jack Silver*
Affiliation:
University of California, Berkeley
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Extract

If X is a set, [Χ] ω will denote the set of countably infinite subsets of X. ω is the set of natural numbers. If S is a subset of [ω] ω , we shall say that S is Ramsey if there is some infinite subset X of ω such that either [Χ] ω ⊆ S or [Χ] ω ∩ S = 0. Dana Scott (unpublished) has asked which sets, in terms of logical complexity, are Ramsey.

The principal theorem of this paper is: Every Σ 1 1 (i.e., analytic) subset of [ω] ω is Ramsey (for the Σ, Π notations, see Addison [1]). This improves a result of Galvin-Prikry [2] to the effect that every Borel set is Ramsey. Our theorem is essentially optimal because, if the axiom of constructibility is true, then Gödel's Σ2 1 Π2 1 well-ordering of the set of reals [3], having the convenient property that the set of ω-sequences of reals enumerating initial segments is also Σ2 1 ∩ Π2 1, rather directly gives a Σ2 1 ∩ Π2 1 set which is not Ramsey. On the other hand, from the assumption that there is a measurable cardinal we shall derive the conclusion that every Σ 2 1 (i.e., PCA) is Ramsey. Also, we shall explore the connection between Martin's axiom and the Ramsey property.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 35 , Issue 1 , March 1970 , pp. 60 - 64
Copyright
Copyright © Association for Symbolic Logic 1970

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Footnotes

1

This work was partially supported by NSF contracts at the University of California, Berkeley.

References

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