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SOME CONSEQUENCES OF ${\mathrm {TD}}$ AND ${\mathrm {sTD}}$

Published online by Cambridge University Press:  15 May 2023

YINHE PENG
Affiliation:
ACADEMY OF MATHEMATICS AND SYSTEMS SCIENCE CHINESE ACADEMY OF SCIENCES EAST ZHONG GUAN CUN ROAD NO. 55 BEIJING 100190, CHINA E-mail: pengyinhe@amss.ac.cn
LIUZHEN WU
Affiliation:
HLM, ACADEMY OF MATHEMATICS AND SYSTEMS SCIENCE CHINESE ACADEMY OF SCIENCES EAST ZHONG GUAN CUN ROAD NO. 55 BEIJING 100190, CHINA E-mail: lzwu@math.ac.cn
LIANG YU*
Affiliation:
DEPARTMENT OF MATHEMATICS NANJING UNIVERSITY NANJING, JIANGSU 210093 PEOPLE’S REPUBLIC OF CHINA

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. (1) ${\mathrm {ZF}}+{\mathrm {TD}}$ implies $\mathrm {wDC}_{\mathbb {R}}$—a weaker version of $\mathrm {DC}_{\mathbb {R}}$.

  2. (2) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every set of reals is measurable and has Baire property.

  3. (3) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every uncountable set of reals has a perfect subset.

  4. (4) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that for every set of reals A and every $\epsilon>0$:

    1. (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.

    2. (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.

Type
Article
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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