Some Consequences of And

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.