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  1.  8
    Reductions Between Types of Numberings.Ian Herbert, Sanjay Jain, Steffen Lempp, Manat Mustafa & Frank Stephan - 2019 - Annals of Pure and Applied Logic 170 (12):102716.
    This paper considers reductions between types of numberings; these reductions preserve the Rogers Semilattice of the numberings reduced and also preserve the number of minimal and positive degrees in their semilattice. It is shown how to use these reductions to simplify some constructions of specific semilattices. Furthermore, it is shown that for the basic types of numberings, one can reduce the left-r.e. numberings to the r.e. numberings and the k-r.e. numberings to the k+1-r.e. numberings; all further reductions are obtained by (...)
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  2.  56
    Skepticism About Reasoning.Sherrilyn Roush, Kelty Allen & Ian Herbert - 2012 - In Gillian Russell & Greg Restall (eds.), New Waves in Philosophy of Science. pp. 112-141.
    Less discussed than Hume’s skepticism about what grounds there could be for projecting empirical hypotheses is his concern with a skeptical regress that he thought threatened to extinguish any belief when we reflect that our reasoning is not perfect. The root of the problem is the fact that a reflection about our reasoning is itself a piece of reasoning. If each reflection is negative and undermining, does that not give us a diminution of our original belief to nothing? It requires (...)
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  3.  4
    A Perfect Set of Reals with Finite Self-Information.Ian Herbert - 2013 - Journal of Symbolic Logic 78 (4):1229-1246.
    We examine a definition of the mutual information of two reals proposed by Levin in [5]. The mutual information iswhereK is the prefix-free Kolmogorov complexity. A realAis said to have finite self-information ifI is finite. We give a construction for a perfect Π10class of reals with this property, which settles some open questions posed by Hirschfeldt and Weber. The construction produces a perfect set of reals withK≤+KA+f for any given Δ20fwith a particularly nice approximation and for a specific choice of (...)
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  4.  1
    On Reals with -Bounded Complexity and Compressive Power.Ian Herbert - 2016 - Journal of Symbolic Logic 81 (3):833-855.
    The Kolmogorov complexity of a finite binary string is the length of the shortest description of the string. This gives rise to some ‘standard’ lowness notions for reals: A isK-trivial if its initial segments have the lowest possible complexity and A is low forKif using A as an oracle does not decrease the complexity of strings by more than a constant factor. We weaken these notions by requiring the defining inequalities to hold only up to all${\rm{\Delta }}_2^0$orders, and call the (...)
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