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Indifferent sets for genericity

Published online by Cambridge University Press:  12 March 2014

Adam R. Day
Adam R. Day*
Affiliation:
School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, Wellington, New Zealand, E-mail: adam.day@math.berkeley.edu
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Abstract

This paper investigates indifferent sets for comeager classes in Cantor space focusing of the class of all 1-generic sets and the class of all weakly 1-generic sets. Jockusch and Posner showed that there exist 1-generic sets that have indifferent sets [10]. Figueira, Miller and Nies have studied indifferent sets for randomness and other notions [7]. We show that any comeager class in Cantor space contains a comeager class with a universal indifferent set. A forcing construction is used to show that any 1-generic set, or weakly 1-generic set, has an indifferent set. Such an indifferent set can by computed by any set in which bounds the (weakly) 1-generic. We show by approximation arguments that some, but not all, 1-generic sets can compute an indifferent set for themselves. We show that all weakly 1-generic sets can compute an indifferent set for themselves. Additional results on indifferent sets, including one of Miller, and two of Fitzgerald, are presented.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 1 , March 2013 , pp. 113 - 138
Copyright
Copyright © Association for Symbolic Logic 2013

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References

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