Some consequences of $\mathrm{TD}$ and $\mathrm{sTD}$

Abstract

Strongly Turing determinacy, or $\mathrm{sTD}$, says that for any set $A$ of reals, if $\forall x\exists y{\ge }_{T}x(y\in A)$, then there is a pointed set $P\subseteq A$. We prove the following consequences of Turing determinacy ($\mathrm{TD}$) and $\mathrm{sTD}$: (1). $\mathrm{ZF}+\mathrm{TD}$ implies weakly dependent choice ($\mathrm{wDC}$). (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 any set of reals $A$ and any $\epsilon >0$, (a) there is a closed set $F\subseteq A$ so that $\mathrm{Di}{\mathrm{m}}_{\mathrm{H}}(F)\ge \mathrm{Di}{\mathrm{m}}_{\mathrm{H}}(A)-\epsilon$. (b) there is a closed set $F\subseteq A$ so that $\mathrm{Di}{\mathrm{m}}_{\mathrm{P}}(F)\ge \mathrm{Di}{\mathrm{m}}_{\mathrm{P}}(A)-\epsilon$.

Better version