Harrington's Solution to McLaughlin's Conjecture and Non-uniform Self-moduli

Abstract

While much work has been done to characterize the Turing degrees computing members of various collections of fast growing functions, much less has been done to characterize the rate of growth necessary to compute particular degrees. Prior work has shown that every degree computed by all sufficiently fast growing functions is uniformly computed by all sufficiently fast growing functions. We show that the rate of growth sufficient for a function to uniformly compute a given Turing degree can be separated by an arbitrary number of jumps from the rate of growth that suffices for a function to non-uniformly compute the degree. These results use the unpublished method Harrington developed to answer McLaughlin's conjecture so we begin the paper with a rigorous presentation of the approach Harrington sketched in his handwritten notes on the conjecture. We also provide proofs for the important computability theoretic results Harrington noted were corollaries of this approach. In particular we provide the first published proof of Harrington's result that there is an effectively given sequence of {\Pi}^0_1 singletons that are Low_{\alpha} none of which is computable in the effective join of the {\alpha} jumps of the others for every computable ordinal {\alpha}.