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2008 Immunity and Hyperimmunity for Sets of Minimal Indices
Frank Stephan, Jason Teutsch
Notre Dame J. Formal Logic 49(2): 107-125 (2008). DOI: 10.1215/00294527-2008-001


We extend Meyer's 1972 investigation of sets of minimal indices. Blum showed that minimal index sets are immune, and we show that they are also immune against high levels of the arithmetic hierarchy. We give optimal immunity results for sets of minimal indices with respect to the arithmetic hierarchy, and we illustrate with an intuitive example that immunity is not simply a refinement of arithmetic complexity. Of particular note here are the fact that there are three minimal index sets located in Π3 − Σ3 with distinct levels of immunity and that certain immunity properties depend on the choice of underlying acceptable numbering. We show that minimal index sets are never hyperimmune; however, they can be immune against the arithmetic sets. Lastly, we investigate Turing degrees for sets of random strings defined with respect to Bagchi's size-function s.


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Frank Stephan. Jason Teutsch. "Immunity and Hyperimmunity for Sets of Minimal Indices." Notre Dame J. Formal Logic 49 (2) 107 - 125, 2008.


Published: 2008
First available in Project Euclid: 15 May 2008

zbMATH: 1142.03024
MathSciNet: MR2402035
Digital Object Identifier: 10.1215/00294527-2008-001

Primary: 03D28
Secondary: 03D45

Keywords: Goedel numberings , hyperimmune sets , immune sets , Kolmogorov numberings , sets of minimal indices , sets of random strings

Rights: Copyright © 2008 University of Notre Dame


Vol.49 • No. 2 • 2008
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