Hyperarithmetically encodable sets
Author:
Robert M. Solovay
Journal:
Trans. Amer. Math. Soc. 239 (1978), 99122
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 be the closure ordinal of a universal inductive definition. Then A is hyperarithmetically encodable iff it is constructible before stage . We also prove an effective version of the GalvinPrikry 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
DOI:
http://dx.doi.org/10.1090/S00029947197804911037
PII:
S 00029947(1978)04911037
Keywords:
Hyperarithmetically encodable set,
Ramsey set,
selective ultrafilter,
inductive definition,
Mathias forcing,
GalvinPrikry theorem
Article copyright:
© Copyright 1978
American Mathematical Society
