اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدSome consequences of $mathrm{TD}$ and $mathrm{sTD}$
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Strongly Turing determinacy, or $mathrm{sTD}$, says that for any set $A$ of reals, if $forall xexists ygeq_T x (yin A)$, then there is a pointed set $Psubseteq A$. 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 $epsilon>0$, (a) there is a closed set $Fsubseteq A$ so that $mathrm{Dim_H}(F)geq mathrm{Dim_H}(A)-epsilon$. (b) there is a closed set $Fsubseteq A$ so that $mathrm{Dim_P}(F)geq mathrm{Dim_P}(A)-epsilon$.
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