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SOME CONSEQUENCES OF
${\mathrm {TD}}$ AND
${\mathrm {sTD}}$
Published online by Cambridge University Press: 15 May 2023
Abstract
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 set of reals is measurable and has Baire property.
(3)
${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every uncountable set of reals has a perfect subset.
(4)
${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that for every set of reals A and every
$\epsilon>0$:
(a) There is a closed set
$F\subseteq A$ such that
$\mathrm {Dim_H}(F)\geq \mathrm {Dim_H}(A)-\epsilon $, where
$\mathrm {Dim_H}$ is the Hausdorff dimension.
(b) There is a closed set
$F\subseteq A$ such that
$\mathrm {Dim_P}(F)\geq \mathrm {Dim_P}(A)-\epsilon $, where
$\mathrm {Dim_P}$ is the packing dimension.
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- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
References
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