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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.

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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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