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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