Notre Dame Journal of Formal Logic

Immunity and Hyperimmunity for Sets of Minimal Indices

Frank Stephan and Jason Teutsch


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.

Article information

Notre Dame J. Formal Logic Volume 49, Number 2 (2008), 107-125.

First available in Project Euclid: 15 May 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03D28: Other Turing degree structures
Secondary: 03D45: Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55]

sets of minimal indices sets of random strings immune sets hyperimmune sets Goedel numberings Kolmogorov numberings


Stephan, Frank; Teutsch, Jason. Immunity and Hyperimmunity for Sets of Minimal Indices. Notre Dame J. Formal Logic 49 (2008), no. 2, 107--125. doi:10.1215/00294527-2008-001.

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