Year: 2021
Type: preprint
Abstract: Strongly Turing determinacy, or $\mathrm{sTD}$, says that for any 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 $\mathrm{sTD}$: (1). $\mathrm{ZF+TD}$ implies weakly dependent choice ($\mathrm{wDC}$). (2). $\mathrm{ZF+sTD}$ implies that every set of reals is measurable and has Baire property. (3). $\mathrm{ZF+sTD}$ implies that every uncountable set of reals has a perfect subset. (4). $\mathrm{ZF+sTD}$ implies that for any set of reals $A$ and any $ε>0$, (a) there is a closed set $F\subseteq A$ so that $\mathrm{Dim_H}(F)\geq \mathrm{Dim_H}(A)-ε$. (b) there is a closed set $F\subseteq A$ so that $\mathrm{Dim_P}(F)\geq \mathrm{Dim_P}(A)-ε$.
Source: arXiv (Cornell University)
Authors: Peng, Yinhe, Wu, Liuzhen, Yu, Liang
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3184570671