Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Hyperarithmetically encodable sets

Author: Robert M. Solovay
Journal: Trans. Amer. Math. Soc. 239 (1978), 99-122
MSC: Primary 02F35; Secondary 02F27, 02F30, 02K99
MathSciNet review: 0491103
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Abstract: We say that a set of integers, A, is hyperarithmetically (recursively) encodable, if every infinite set of integers X contains an infinite subset Y in which A is hyperarithmetical (recursive). We show that the recursively encodable sets are precisely the hyperarithmetic sets. Let $ \sigma $ be the closure ordinal of a universal $ \Sigma _1^1$ inductive definition. Then A is hyperarithmetically encodable iff it is constructible before stage $ \sigma $.

We also prove an effective version of the Galvin-Prikry results that open sets, and more generally Borel sets, are Ramsey, and in the case of open sets prove that our improvement is optimal.

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Keywords: Hyperarithmetically encodable set, Ramsey set, selective ultrafilter, inductive definition, Mathias forcing, Galvin-Prikry theorem
Article copyright: © Copyright 1978 American Mathematical Society