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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Hyperarithmetically encodable sets
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by Robert M. Solovay PDF
Trans. Amer. Math. Soc. 239 (1978), 99-122 Request permission

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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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 239 (1978), 99-122
  • MSC: Primary 02F35; Secondary 02F27, 02F30, 02K99
  • DOI: https://doi.org/10.1090/S0002-9947-1978-0491103-7
  • MathSciNet review: 0491103