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  1. Some Consequences of And.Yinhe Peng, W. U. Liuzhen & Y. U. Liang - 2023 - Journal of Symbolic Logic 88 (4):1573-1589.
    Strong Turing Determinacy, or ${\mathrm {sTD}}$, is the statement that for every set A of reals, if $\forall x\exists y\geq _T x (y\in A)$, then there is a pointed set $P\subseteq A$. We prove the following consequences of Turing Determinacy ( ${\mathrm {TD}}$ ) and ${\mathrm {sTD}}$ over ${\mathrm {ZF}}$ —the Zermelo–Fraenkel axiomatic set theory without the Axiom of Choice: (1) ${\mathrm {ZF}}+{\mathrm {TD}}$ implies $\mathrm {wDC}_{\mathbb {R}}$ —a weaker version of $\mathrm {DC}_{\mathbb {R}}$.(2) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every (...)
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  • 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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