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

TL;DR

This paper explores the implications of Turing determinacy and strongly Turing determinacy, showing they lead to significant regularity properties of sets of reals, including measurability, the Baire property, and the existence of perfect subsets.

Contribution

It establishes new consequences of $ ext{TD}$ and $ ext{sTD}$, such as measurability, Baire property, and perfect subsets for all uncountable reals, expanding understanding of their foundational impact.

Findings

ZF+TD implies weakly dependent choice.

ZF+sTD implies all sets of reals are measurable and have Baire property.

ZF+sTD implies every uncountable set of reals contains a perfect subset.

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 $\epsilon>0$, (a) there is a closed set $F\subseteq A$ so that $\mathrm{Dim_H}(F)\geq \mathrm{Dim_H}(A)-\epsilon$. (b) there is a closed set $F\subseteq A$ so that $\mathrm{Dim_P}(F)\geq \mathrm{Dim_P}(A)-\epsilon$.