SOME CONSEQUENCES OF ${\mathrm {TD}}$ AND ${\mathrm {sTD}}$

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. (1) ${\mathrm {ZF}}+{\mathrm {TD}}$ implies $\mathrm {wDC}_{\mathbb {R}}$—a weaker version of $\mathrm {DC}_{\mathbb {R}}$.

2. (2) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every set of reals is measurable and has Baire property.

3. (3) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every uncountable set of reals has a perfect subset.

4. (4) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that for every set of reals A and every $\epsilon>0$:

   1. (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.

   2. (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.

1 Introduction

1.1 Turing Determinacy and strong Turing Determinacy

Turing reduction $\leq _T$ is a quasi-order over reals. $x\leq _T y$ , or x is Turing reducible to y, means x can be computed by a Turing machine with oracle y. The reduction naturally induces an equivalence relation $\equiv _T$ . Given a real x, its corresponded Turing degree ${\mathbf {x}}$ is a set of reals defined as $\{y\mid y\equiv _T x\}$ . We say ${\mathbf {x}}\leq \mathbf {y}$ if $x\leq _T y$ . We use $\mathcal {D}$ to denote the set of Turing degrees. An upper cone of Turing degrees is a set of form $\{\mathbf {y}\mid \mathbf {y}\geq \mathbf {x} \}$ for some $\mathbf {x}$ .

We say that a perfect set P is pointed if there is a perfect tree $T\subseteq 2^{<\omega }$ such that $[T]=P$ and for any $x\in P$ , $T\leq _T x$ , where $[T]=\{x\in 2^{\omega }\mid \forall n (x{\upharpoonright} n \in T)\}$ .

Definition 1.1.

• • Turing Determinacy, or ${\mathrm {TD}}$ , is the statement that for every set A of Turing degrees, either A or $\mathcal {D}\setminus A$ contains an upper cone of Turing degrees.

• • 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 such that $P\subseteq A$ .

Clearly ${\mathrm {sTD}}$ implies ${\mathrm {TD}}$ . Martin proves the following famous theorem, which partly justifies why Turing degrees are important to set theory.

Theorem 1.2 (Martin [16])

Over ${\mathrm {ZF}}$ , Axiom of Determinacy, or ${\mathrm {AD}}$ , implies ${\mathrm {sTD}}$ and so ${\mathrm {TD}}$ .

Definition 1.3.

• • Countable choice axiom for sets of reals, or ${\mathrm {CC}_{\mathbb {R}}}$ , is the statement that for every countable sequence $\{A_n\}_{n\in \omega }$ of nonempty sets of reals, there is a function $f:\omega \to \mathbb {R}$ such that for every n, $f(n)\in A_n$ .

• • Dependent choice axiom for sets of reals, or ${\mathrm {DC}_{\mathbb {R}}}$ , is the statement that for every binary relation R over reals such that $\forall x \exists y R(x,y)$ , there is a function $f:\omega \to \mathbb {R}$ such that for every n, $R(f(n), f(n+1))$ .

Turing Determinacy is an important and very useful consequence of ${\mathrm {AD}}$ . In many situations, ${\mathrm {TD}}$ seems sufficient to be used to prove set theory theorems. The following theorem partly justifies this phenomenon. Actually Woodin shows that under ${\mathrm {ZF}}+{\mathrm {TD}}+{\mathrm {DC}_{\mathbb {R}}}$ , every Suslin set is determined (see [28, Theorem 1.2]).

Theorem 1.4 (Woodin [28, Theorem 1.1])

Assume ${\mathrm {ZF}}+V=L(\mathbb {R})+{\mathrm {DC}_{\mathbb {R}}}$ . ${\mathrm {AD}}$ is equivalent to ${\mathrm {TD}}$ .

Moreover, as we shall see in this paper, the consequences of ${\mathrm {TD}}$ (and ${\mathrm {sTD}}$ ) may be proved in a uniform way, unlike the proofs that assume ${\mathrm {AD}}$ , which often require a very genius, tricky, and case-by-case design of games.

The first result in this paper concerns the relationship between ${\mathrm {AD}}$ and Axiom of Choice, or $\mathrm {AC}$ . Although ${\mathrm {AD}}$ contradicts $\mathrm {AC}$ , Mycielski proves the following theorem.

Theorem 1.5 (Mycielski [18])

Over ${\mathrm {ZF}}$ , ${\mathrm {AD}}$ implies ${\mathrm {CC}_{\mathbb {R}}}$ .

The question if ${\mathrm {AD}}$ implies ${\mathrm {DC}_{\mathbb {R}}}$ remains open for a long time.

Question 1.6 (Solovay [26])

Over ${\mathrm {ZF}}$ , does ${\mathrm {AD}}$ imply ${\mathrm {DC}_{\mathbb {R}}}$ ?

Kechris proves the following result.

Theorem 1.7 (Kechris [12])

Assume ${\mathrm {ZF}}+V=L(\mathbb {R})$ . ${\mathrm {AD}}$ implies ${\mathrm {DC}_{\mathbb {R}}}$ .

It is unknown whether the assumption $V=L(\mathbb {R})$ can be removed. Recently, the following “unconditional” result is proved.

Theorem 1.8 (Peng and Yu [20])

Over ${\mathrm {ZF}}$ , ${\mathrm {TD}}$ implies ${\mathrm {CC}_{\mathbb {R}}}$ .

We will use ${\mathrm {CC}_{\mathbb {R}}}$ throughout the paper even without mentioning it.

The first result in this paper is a partial solution to Question 1.6. We prove in Theorem 4.3 that ${\mathrm {ZF}}+{\mathrm {TD}}$ implies $\mathrm {wDC}_{\mathbb {R}}$ , a weaker version of ${\mathrm {DC}_{\mathbb {R}}}$ (for the definition of $\mathrm {wDC}_{\mathbb {R}}$ , see Definition 4.1).

The second result in this paper is about the regularity properties of sets of reals. Although ${\mathrm {TD}}$ seems unlikely as strong as ${\mathrm {AD}}$ , a natural question is whether ${\mathrm {TD}}$ is as “useful” as ${\mathrm {AD}}$ . Sami initiated this project by proving (in [23]) that ${\mathrm {ZF}}+{\mathrm {TD}}$ implies $\mathrm {CH}$ , the continuum hypothesis. But it seems a rather difficult (and long-standing) question whether ${\mathrm {ZF}}+{\mathrm {TD}}(+\mathrm {DC})$ implies regularity properties for sets of reals. A progress is made in [8] by showing that the perfect set property for all Turing invariant sets of reals implies the perfect set property for all sets of reals. A partial answer (Theorem 5.1) to this question is due to Woodin and remains unpublished (email communication between Woodin and Yu in April 2021). In this paper, we give another proof of Theorem 5.1 that strong Turing Determinacy, ${\mathrm {sTD}}$ –a stronger version of ${\mathrm {TD}}$ —implies the regularity properties for sets of reals via a recursion theoretical method that is different from Woodin’s.

A basis for a class $\mathscr {C}$ of linearly ordered sets is a collection $\mathscr {B}\subseteq \mathscr {C}$ such that for each $L_1\in \mathscr {C}$ , there is an $L_2\in \mathscr {B}$ such that $L_2$ is isomorphic to a subset of $L_1$ . Investigating basis for linear ordering is a very active area in set theory today. For example, Moore [17] proves that under proper forcing axioms, $\mathrm {PFA}$ , a five-element basis for uncountable linear orders exists. But it seems that basis theorems for linear orderings under ${\mathrm {AD}}$ remain untouched. In this paper, we prove a basis theorem for linear orderings over $\mathbb {R}$ under the assumption ${\mathrm {ZF}}+{\mathrm {TD}}+{\mathrm {DC}_{\mathbb {R}}}+\mbox {"every uncountable set of reals has a perfect subset"}$ by showing that for every linear order $\leq _L$ over $\mathbb {R}$ , there is an order-preserving embedding from $(2^{\omega },\leq )$ to $(\mathbb {R},\leq _L)$ . In other words, $\{(2^{\omega },\leq )\}$ is a basis for $\{(\mathbb {R},\leq _L)\mid \leq _L \mbox { is a linear ordering over } \mathbb {R}\}.$

The last result in this paper is an application of recursion theory to fractal geometry theory. Besicovitch and Davis [1, 5] prove that for every analytic set, its Hausdorff dimension can be approximated arbitrarily close by the Hausdorff dimension of its closed subsets. Joyce and Preiss [11] prove a similar result for packing dimension. Recently Slaman [25] proves that both Besicovitch–Davis and Joyce–Preiss theorems fail for some $\Pi ^1_1$ -set under the assumption $V=L$ . However, we prove in Theorem 6.5 that both theorems hold for arbitrary sets of reals over ${\mathrm {ZF}}+{\mathrm {sTD}}$ . So the phenomenon can be viewed as a new regularity property for the sets of reals. This weakens the assumption of the following result in [3].

Theorem 1.9 (Crone, Fishman, and Jackson [3])

Assume ${\mathrm {ZF}}+{\mathrm {AD}}+{\mathrm {DC}_{\mathbb {R}}}$ . For every set A and $\epsilon>0$ , there is a closed set $F\subseteq A$ such that $\mathrm {Dim_H}(F)\geq \mathrm {Dim_H}(A)- \epsilon $ .

The proof of this theorem is direct and uses some rather deep results from set theory. However, we believe that our proof is simpler and, most importantly, it does not depend on ${\mathrm {DC}_{\mathbb {R}}}$ .

1.2 Point to set principle

Relativization opens a door between recursion theory and other mathematical branches. In recursion theory, for a real x, a relativization to x, roughly speaking, is a way to add prefix x- to every appearance of any notion in the statement. Then if a notion is defined in recursion theory, its relativization is defined naturally. And if a theorem in recursion theory is proved, then its relativization also follows naturally. For example, every continuous function is a recursive function relative to a real, and a Borel set is a hyperarithmeitc set relative to a real. From this point of view, one may apply recursion theory results to analysis.

The “point to set” principle is a more concrete way, by using relativization, to apply recursion theory to other areas of mathematics. Generally speaking, the principle says that a set A having certain property is equivalent to that it contains some special points. Such argument can be dated back to Sacks, who (in [21]) gave a recursion theoretical proof of the classical result that every analytic set is measurable. For one more example, given a relativizable algorithmic randomness notion $\Gamma $ (such as Martin-Löf and Schnorr), we have the following fact.

Fact 1.10. Assume ${\mathrm {ZF}}+{\mathrm {CC}_{\mathbb {R}}}$ . A set $A\subseteq \mathbb {R}$ is null if and only if there is some real x such that no $\Gamma (x)$ -random real is in A.

So a set A is not null is equivalent to say that for every real x, there is a $\Gamma (x)$ -random real in A. One may also replace randomness with genericity and obtain the corresponding results. In this paper, we apply some recent results in recursion theory and algorithmic randomness theory to descriptive set theory and fractal geometry theory. Especially some deep results concerning the lowness properties for various recursion theory notations turned to be crucial to our proof. The so-called “lowness properties” is a kind of property preserving some algorithmic property that was considered as very unique in algorithmic randomness theory. For example, a real x is low for Turing jump (or just low) if $x'\equiv _T \emptyset '$ , and a real x is called low for Schnorr random (for the definition of Schnorr randomness, see the paragraphs below Theorem 4.3) if every Schnorr random real is Schnorr random relative to x, etc. Ironically, different from the “slowdown” properties of themselves, these notions will be used to prove some “speedup” results. We expect to see more such applications in the near future.

In [15], a specific theorem (i.e., Theorem 6.6) is called the “point to set” principle. It can be viewed as a dual principle to the randomness in geometric measure theory.

We organize the paper as follows. In Section 2, we give some terminologies and notions. In Section 3, we sketch a recursion theoretical reformulation of Sami’s proof that ${\mathrm {ZF}}+{\mathrm {TD}}$ implies $\mathrm {CH}$ . The result will be used in Section 5. In Section 4, we prove $\mathrm {wDC}_{\mathbb {R}}$ within ${\mathrm {ZF}}+{\mathrm {TD}}$ . In Section 5, we prove that ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies regularity properties for sets of reals. In the same section, we also prove a basis theorem for linear orderings over sets of reals within ${\mathrm {ZF}}+{\mathrm {TD}}+{\mathrm {DC}_{\mathbb {R}}}+\mbox {"every uncountable set of reals has a perfect subset."}$ In Section 6, we prove that the Besicovitch–Davis theorem holds for every set of reals within ${\mathrm {ZF}}+{\mathrm {sTD}}$ .

2 Terminologies and notions

We assume that readers have some knowledge of descriptive set theory and recursion theory. The major references are [2, 6, 10, 14, 19, 22].

2.1 Set theory

We assume that readers have some knowledge of axiomatic set theory. ${\mathrm {ZF}}$ is the Zermelo–Fraenkel axiom system. ${\mathrm {AD}}$ is the axiom of Determinacy.

When we say that $T\subseteq 2^{<\omega }$ is a tree, we mean that T is a tree without dead nodes. $[T]$ is the collection of infinite paths through T. Given any $x\in \omega ^{\omega }$ and natural number n, we use $x{\upharpoonright} n$ to denote an initial segment of x with length n. In other words, $x{\upharpoonright} n$ is a finite string $\sigma \in \omega ^{<\omega }$ of length n such that for any $i<n$ , $\sigma (i)=x(i)$ .

2.2 Recursion theory

We use $\leq _T$ to denote Turing reduction and $\leq _h$ to denote hyperarithmetic reduction. We use $\Phi ^x$ to denote a Turing machine with oracle x. Sometimes we also say that $\Phi ^x$ is a recursive functional. We fix an effective enumeration $\{\Phi _e^x\}_{e\in \omega }$ of recursive functionals.

We use Kleene’s $\mathcal {O}$ and we write $\omega _1^{\mathrm {CK}}$ for the least non-recursive ordinal, $\omega _1^x$ for the least ordinal not recursive in x.

We say a set A ranges Turing degrees cofinally if for every real x, there is some $y\geq _T x$ in A. We use $x'$ to denote the Turing jump relative to x. More generally, if $\alpha <\omega _1^x$ , then $x^{(\alpha )}$ is that $\alpha $ -th Turing jump of x. Then, $x\leq _h y$ if $x\leq _T y^{(\alpha )}$ for some $\alpha <\omega _1^y$ .

The following fact is folklore and a sketched proof can be found in [20].

Lemma 2.1. Assume ${\mathrm {ZF}}$ . For any Turing degree ${\mathbf {x}}$ , there is a family of Turing degrees $\{{\mathbf {y}}_r\mid r\in \mathbb {R}\}$ satisfying the following properties $:$

1. (1) For any $r\in \mathbb {R}$ , ${\mathbf {x}}<{\mathbf {y}}_r$ .

2. (2) For any $r_0\neq r_1\in \mathbb {R}$ and ${\mathbf {z}}<{\mathbf {y}}_{r_0},{\mathbf {y}}_{r_1}$ , we have that ${\mathbf {z}}\leq {\mathbf {x}}$ .

3. (3) For any ${\mathbf {z}}\geq {\mathbf {x}}"$ , the Turing double jump of ${\mathbf {x}}$ , there is an infinite set $C_{{\mathbf {z}}}\subset \mathbb {R}$ such that ${\mathbf {y}}_r"={\mathbf {z}}$ for any $r\in C_{{\mathbf {z}}}$ .

3 On Sami’s theorem

Theorem 3.1 (Sami [23])

${\mathrm {ZF}}+{\mathrm {TD}}+\mathrm {DC}$ proves $\mathrm {CH}$ .

In this section, we sketch a recursion theoretical proof of Theorem 3.1 to show that $\mathrm {DC}$ can be removed from the assumption, which was also observed by Sami. So we can give another proof of the following result.

Proposition 3.2 (Sami, email communication between Sami and Yu in June 2021)

${\mathrm {ZF}}+{\mathrm {TD}}$ proves $\mathrm {CH}$ .

Proof Given an uncountable set $A\subseteq \mathbb {R}$ , by Lemma 2.1, for any real x, there is a real $y>_T x$ such that there is some real $r\in A$ Turing below $y"$ but not below y. So, by ${\mathrm {TD}}$ , there is some real $z_0$ such that for any $y\geq _T z_0$ , there is some real $r\in A$ Turing below $y"$ but not below y.

Now it is simple to construct a $\Sigma ^1_1(z_0)$ set B so that:

1. (i) For any $y\leq _h z_0$ and $x\in B$ , we have that $y\leq _T x$ .

2. (ii) For any $x_0\neq x_1\in B$ , if $y\leq _h x_0, x_1$ , then $y\leq _h z_0$ .

To see the existence of such B, first note that the set $\{y\mid \forall r\leq _h z_0(r\leq _T y)\}$ is an uncountable $\Sigma ^1_1(z_0)$ -set. Then one may construct a perfect set $P\subseteq B$ so that any two different members from P form a minimal pair over $z_0$ in the hyperarithmetic degree sense.

Now for every real $x\in B$ , let $y_x=\Phi ^{x"}_e$ where e is the least n such that $\Phi ^{x"}_n$ is in A and not Turing below x. For any $x_0\neq x_1\in B$ , if $y_{x_0}=y_{x_1}$ , then by (ii), $y_{x_0}=y_{x_1}\leq _h z_0$ . By (i), we have that $y_{x_0}=y_{x_1}\leq _T x_0$ , which is a contradiction.

So $x\mapsto y_x$ is a $1$ – $1$ map from B to A. It is known that every uncountable analytic set has a perfect subset and so A has the same power as $\mathbb {R}$ .

From the proof of Proposition 3.2, we can see the following fact that will be used later. Actually by the remark after the proof of [23, Theorem 1.3], Sami proves that $f_n$ below can be chosen to be continuous. But we only need this weaker version here.

Lemma 3.3 (Sami [23])

Assume ${\mathrm {ZF}}+{\mathrm {TD}}$ . For every uncountable set A of reals, there is a perfect set B of reals and a sequence of arithmetical functions $\{f_n\}_{n\in \omega }$ from B to $\mathbb {R}$ such that $B\subseteq \bigcup _{n\in \omega }f_n^{-1}(A)$ . Moreover, restricted to B, $f_n$ is 1–1 for every n.

Proof Fix an effective enumeration of Turing functional $\{\Phi _n\}_{n\in \omega }$ . Going to a subset, assume that in the proof of Proposition 3.2, B is perfect. Define $f_n: B\to \mathbb {R}$ by

(1) $$ \begin{align} f_n(x)=\left\{ \begin{array}{rcl} \uparrow, & &(\exists m \Phi_n^{x"}(m)\mbox{ is not defined}) \vee( \Phi_n^{x"}\leq_T x),\\[2pt] \Phi_n^{x"}, & & \mathrm{otherwise}.\\ \end{array} \right. \end{align} $$

Clearly $f_n$ is arithmetical for every n. We have that $B\subseteq \bigcup _{n\in \omega }f_n^{-1}(A)$ . Moreover, if $x\in B$ and $f_n(x)$ is defined, then $f_n(x)\leq _T x"\wedge f_n(x)\not \leq _T x$ . Then by the same reason as in the proof of Proposition 3.2, $f_n$ must be 1–1 on B. So $\{f_n\}_{n\in \omega }$ is as required.

4 Weakly dependent choice

Throughout the section, we work within ${\mathrm {ZF}}+{\mathrm {TD}}$ .

Definition 4.1. Weakly dependent choice for sets of reals, or $\mathrm {wDC}_{\mathbb {R}}$ , is the statement that for every binary relation R over $\mathbb {R}$ with the property that the set $\{y\mid R(x,y)\}$ has positive inner measure for any real x, there is a sequence $\{x_n\}_{n\in \omega }$ of reals such that $\forall n R(x_n,x_{n+1})$ .

Remark. $\mathrm {wDC}_{\mathbb {R}}$ is not a consequence of ${\mathrm {ZF}}$ . To see this, let V be a model of ZFC+GCH and $\kappa =\aleph _\omega ^V$ . Let G be $Col(\omega , <\kappa )$ -generic over V. Let $A_n=\kappa ^\omega \cap V[G\cap Col(\omega , <\aleph _n^V)]$ . Let

$$ \begin{align*}M=HOD^{V[G]}_{\bigcup A_n\cup \{\langle A_n: n<\omega\rangle\}}.\end{align*} $$

Then in M:

• • $2^\omega \cap M=2^\omega \cap (\bigcup A_n)$ . [See, e.g., the proof of [24, Theorem 6.69].]

• • $A_n$ is countable.

• • Every countable subset of reals intersects at most finitely many $A_{n+1}\setminus A_n$ ’s.

For every $x\in 2^\omega \cap M$ , let $n(x)$ be the least n such that $x\in A_n$ . Then in M,

$$ \begin{align*}R=\{(x, y)\in 2^\omega\times 2^\omega: n(x)<n(y)\}\end{align*} $$

witnesses the failure of $\mathrm {wDC}_{\mathbb {R}}$ .

Proposition 4.2. $\mathrm {wDC}_{\mathbb {R}}$ does not imply ${\mathrm {CC}_{\mathbb {R}}}$ .

Proof To see this, let $V=L[A]$ for some set $A $ of ordinals, $\mathbb {P}$ be the random forcing over $2^{\omega \times \omega }$ , G be $\mathbb {P}$ -generic over V, and $r_G: \omega \times \omega \rightarrow 2$ be the induced random real. Let $x_n=r_G(n,\cdot ): \omega \rightarrow 2$ for each $n<\omega $ and $X=\{x_n: n<\omega \}$ . Let

$$ \begin{align*}M=HOD^{V[G]}_{\{A\}\cup X\cup \{X\}}.\end{align*} $$

First note that any bijection $\pi : \omega \rightarrow \omega $ will induce a homeomorphism $\tilde {\pi }$ of $2^{\omega \times \omega }$ to itself by: $\tilde {\pi }(x)(n,m)=x(\pi (n),m)$ . Moreover, $\tilde {\pi }$ preserves measure and X. Then the following hold in M.

1. (1) M satisfies $\mathrm {wDC}_{\mathbb {R}}$ . [Let R be a binary relation in M satisfying the assumption of $\mathrm {wDC}_{\mathbb {R}}$ . Find $a\in V$ , $x_0,\ldots ,x_{m-1}\in X$ and formula $\varphi $ such that $x R y$ iff $V[G]\vDash \varphi (x,y, a,x_0,\ldots ,x_{m-1}, X).$ We may assume that there is no occurrence of $x_i$ in $\varphi $ since, e.g., we may view $L[A, x_0,\ldots ,x_{m-1}]$ as the ground model and $r_G\upharpoonright _{[m,\omega )\times \omega }$ as the random real over $L[A, x_0,\ldots ,x_{m-1}]$ . So

   $$ \begin{align*}x R y\text{ iff }V[G]\vDash \varphi(x,y, a, X).\end{align*} $$
   Recall that $\mathbb {R}\cap V$ has full outer measure in $V[G]$ . Consequently, for any $x\in \mathbb {R}\cap V$ , there is some $y\in V$ such that $x R y$ . We will be done if $R\cap (\mathbb {R}\cap V)^2\in V$ . Now it suffices to prove that for any $x, y\in \mathbb {R}\cap V$ , for any $p, q\in \mathbb {P}$ ,
   $$ \begin{align*}p\Vdash \varphi(x,y,a, \dot{X}) \text{ iff } q\Vdash \varphi(x,y,a, \dot{X}).\end{align*} $$
   Suppose towards a contradiction that $p\Vdash \varphi (x,y,a, \dot {X}) \text { and } q\Vdash \neg \varphi (x,y,a, \dot {X}).$ Find finite unions of basis open sets $O_p$ and $O_q$ such that for some n:

   • • $\mu (O_p)=\mu (O_q)=\epsilon $ for some $\epsilon>0$ ;

   • • $\mu (O_p\setminus p)+\mu (O_q\setminus q)<\epsilon ^2$ ;

   • • $O_p=\bigcup _{i<k} O_{\sigma _i}$ and each $\sigma _i: n\times \omega \rightarrow 2$ is a finite partial map;

   • • $O_q=\bigcup _{i<k^*} O_{\tau _i}$ and each $\tau _i: n\times \omega \rightarrow 2$ is a finite partial map.

   Fix a bijection $\pi : \omega \rightarrow \omega $ such that $\pi [[0,n)]\subset [n, \omega )$ . Then

   $$ \begin{align*}\mu(p\cap \tilde{\pi}(q))\geq \mu(O_p\cap \tilde{\pi}[O_q])-\mu(O_p\setminus p)-\mu(\tilde{\pi}[O_q\setminus q])>0.\end{align*} $$
   But recall that $\tilde {\pi }[\dot {X}]=\dot {X}$ . So
   $$ \begin{align*}p\cap \tilde{\pi}(q)\Vdash \varphi(x,y,a, \dot{X})\wedge \neg \varphi(x,y,a, \dot{X}).\end{align*} $$
   A contradiction.]

2. (2) There is no injection from $\omega $ to X. [Suppose otherwise, f is an injection. Then for some $a\in V$ , $x_0,\ldots ,x_{m-1}\in X$ and formula $\varphi $ , for any $(n, x)\in \omega \times X$ , $f(n)=x$ iff $V[G]\vDash \varphi (n,x,a,x_0,\ldots ,x_{m-1}, X).$ Choose n such that $f(n)=x_k\notin \{x_0,\ldots ,x_{m-1}\}$ and $p\in G$ such that $p\Vdash \varphi (n,\dot {x}_k,a,\dot {x}_0,\ldots ,\dot {x}_{m-1}, \dot {X}).$ It is straightforward to find a bijection $\pi : \omega \rightarrow \omega $ such that $\pi $ is identity on $\{0,\ldots , m-1\}$ , $\pi (k)\neq k$ and $\mu (p\cap \tilde {\pi }[p])>0$ . But then $p\cap \tilde {\pi }[p]\Vdash \dot {x}_k=\dot {f}(n)=\dot {x}_{\pi (k)}$ . A contradiction.]

So M does not satisfy ${\mathrm {CC}_{\mathbb {R}}}$ . To see this, find a sequence of disjoint rational intervals $\langle I_n: n<\omega \rangle $ such that $I_n\cap X\neq \emptyset $ for each n. Then by (2), $\langle I_n\cap X: n<\omega \rangle $ admits no choice function.

If we use Cohen forcing instead of random forcing in the above argument, then we conclude that the category version of $\mathrm {wDC}_{\mathbb {R}}$ does not imply ${\mathrm {CC}_{\mathbb {R}}}$ . But we do not know if ${\mathrm {CC}_{\mathbb {R}}}$ implies $\mathrm {wDC}_{\mathbb {R}}$ .

Theorem 4.3. ${\mathrm {ZF}}+{\mathrm {TD}}$ implies $\mathrm {wDC}_{\mathbb {R}}$ .

We remark that if “having positive inner measure” is replaced by having Baire property and non-meager in the definition of $\mathrm {wDC}_{\mathbb {R}}$ , then the theorem still holds.

A Schnorr test relative to x is an x-recursive sequence of x-recursive open sets $\{V_{n}\}_{n\in \omega }$ such that $\forall n \mu (V_n)=2^{-n}$ . A real r is called x-Schnorr random if $r\not \in \bigcap _{n\in \omega } V_n$ for any Schnorr test $\{V_{n}\}_{n\in \omega }$ relative to x. If x is recursive, then we simply use Schnorr randomness instead of x-Schnorr randomness. It is not difficult to see that there is a Schnorr random $r\leq _T \emptyset '$ . And it is clear from the definition that if r is x-Schnorr random and $z\leq _T x$ , then r is also z-Schnorr random. Also note that if $x\geq _T \emptyset '$ and r is x-Schnorr random, then x is Turing incomparable with r.

A real x is called low for Schnorr random if every Schnorr random real is Schnorr random relative to x. The following theorem, which was proved by Sacks forcing, is due to Terwijn and Zambella.

Theorem 4.4 (Terwijn and Zambella [27])

For every real $y\geq _T \emptyset "$ , there is a real x low for Schnorr random such that $x"\equiv _T y$ .

Figure 1 $R(r_n,r_{n+1})$ .

Proof of Theorem 4.3

Fix a binary relation R as stated in $\mathrm {wDC}_{\mathbb {R}}$ . To prove $\mathrm {wDC}_{\mathbb {R}}$ , we may assume that for any real x, the set $R_x=\{y\mid R(x,y)\}$ is upward closed under Turing reduction (and so $R_x$ is co-null for any x). I.e., for any y and z, if $y\leq _T z$ and $y\in R_x$ , then $z\in R_x$ . To see this, we may define a new relation $\tilde {R}$ so that $\tilde {R}(x,y)$ if and only if $\forall z_0\leq _T x \exists z_1\leq _T y \ R(z_0,z_1)$ . Then for every real x, the set $\tilde {R}_x=\{y\mid \tilde {R}(x,y)\}$ is upward closed under Turing reduction and has positive measure, and so co-null. Moreover, if there is a sequence $\{y_n\}_{n\in \omega }$ such that $\forall n\tilde {R}(y_n,y_{n+1})$ . Then we build a sequence $\{x_n\}_{n\in \omega }$ so that $\forall nR(x_n,x_{n+1})$ as follows.

First let $x_0= y_0$ . By the definition of $\tilde {R}$ , we may choose the least $m_1$ such that $\Phi _{m_1}^{y_1}$ is defined and $R(x_0, \Phi _{m_1}^{y_1})$ . Let $x_1=\Phi _{m_1}^{y_1}$ . Generally, if $x_n$ is defined, then $x_n\leq _T y_n$ . So by the definition of $\tilde {R}$ , we may choose the least index $m_{n+1}$ such that $\Phi _{m_{n+1}}^{y_{n+1}}$ is defined and $R(x_n, \Phi _{m_{n+1}}^{y_{n+1}})$ . Set $x_{n+1}=\Phi _{m_{n+1}}^{y_{n+1}}$ . Then we have that $\forall n R(x_n,x_{n+1})$ .

Now fix a real z. Note that by the assumption on R, $\bigcap _{y\leq _T z'} R_y$ is co-null. Then by Fact 1.10, there is a real $z_0\geq _T z'$ such that for every $y\leq _T z'$ and every $z_0$ -Schnorr random r, $R(y,r)$ . Also by relativizing Theorem 4.4 to z, there is a real $x>_T z$ low for z-Schnorr random such that $x"\geq _T z_0$ . So for every $y\leq _T z'$ and every $x"$ -Schnorr random r, $R(y,r)$ . Also note that there is a z-Schnorr random, and so x-Schnorr random, real $r\leq _T z'$ . Since $x"\geq _T z'$ , there is some index of Turing functional e such that $\Phi _e^{x"}=z'$ . For every number $e\in \omega $ , define the set

$$ \begin{align*} A_e=\{x\mid \exists r(r\mbox{ is }x\mbox{-Schnorr random }\wedge r\leq_T \Phi_e^{x"})\\ \wedge \forall r_0\leq_T \Phi_e^{x"}\forall r_1(r_1\mbox{ is }x"\mbox{-Schnorr random}\rightarrow R(r_0,r_1))\}. \end{align*} $$

Then by the discussion above, $\bigcup _{e\in \omega }A_e$ ranges Turing degrees cofinally. So there must be some $e_0$ such that $A_{e_0}$ ranges Turing degrees cofinally. By ${\mathrm {TD}}$ , there is some $x_0$ such that every $y\geq _T x_0$ is $\equiv _T$ equivalent to some $y_0$ in $A_{e_0}$ . We may assume that $x_0\in A_{e_0}$ . Recursively in $x_0^{(\omega )}$ , we first find a sequence of reals

$$ \begin{align*}\{y_n\in A_{e_0}\mid n<\omega \wedge y_n\equiv_T x_0^{(2n)}\}.\end{align*} $$

Then find a sequence of reals $\{r_n\}_{n\in \omega }$ so that for every n, $r_n\leq _T \Phi _{e_0}^{y_n"}$ is $y_n\equiv _T x^{(2n)}_0$ -Schnorr random. Note that for every n, $r_n\leq _T \Phi _{e_0}^{y_n"}$ and $r_{n+1}$ is $x^{(2n+2)}_0\equiv _T y_n"$ -Schnorr random (see Figure 1). So by the definition of $A_{e_0}$ , $R(r_n,r_{n+1})$ .

The reason we choose Schnorr randomness, instead of Martin-Löf randomness that is the standard randomness notion, is that every low for Martin-Löf random real is Turing below $\emptyset '$ . So for any real x low for Martin-Löf random, there is no way to make the Turing jumps of x be very high.

5 Regularity properties of sets of reals

In this section, we prove some regularity properties for sets of reals under ${\mathrm {ZF}}+{\mathrm {sTD}} (+{\mathrm {DC}_{\mathbb {R}}})$ . Woodin already considered ${\mathrm {sTD}}$ long time ago. All the results in Sections 5.1 and 5.2 have been known to him (email communication between Woodin and Yu in April 2021).

Theorem 5.1 (Woodin, unpublished)

1. (1) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every set is measurable and has Baire property.

2. (2) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every uncountable set of reals has a perfect subset.

5.1 The proof of part (1)

We only prove that every set is measurable and leave the second part to readers.

It suffices to prove that for any set A, if every measurable subset of A is null, then A must be null. Now suppose, for a contradiction, that every measurable subset of A is null but A is not null. Then, by Fact 1.10 with Schnorr randomness, for every real z, there is a z-Schnorr random real $z_0$ in A. By Theorem 4.4 relative to z, there is a real $x\geq _T z$ low for z-Schnorr random and $x"\geq _T z_0$ .

Now for every $e\in \omega $ , let

$$ \begin{align*}B_e=\{x\mid \Phi_e^{x"}\in A \mbox{ is an }x\mbox{-Schnorr random real}\}.\end{align*} $$

The argument in the previous paragraph shows that $\bigcup _{e\in \omega } B_e$ ranges Turing degrees cofinally. Then there is some $e_0$ such that $B_{e_0}$ ranges Turing degrees cofinally. By ${\mathrm {sTD}}$ , there is a pointed subset $P\subseteq B_{e_0}$ .

Let

$$ \begin{align*}C=\{r\mid \exists x\in P(\Phi_{e_0}^{x"}=r)\}.\end{align*} $$

C is an analytic set and so measurable. Since P is a pointed set, by the definition of $B_{e_0}$ and Fact 1.10 with Schnorr randomness, C is not null. This is absurd.

Remark. Clearly the proof can be localized if we assume ${\mathrm {CC}_{\mathbb {R}}}$ , as pointed out by one of the referees. For example, assuming ${\mathrm {ZF}}+{\mathrm {CC}_{\mathbb {R}}}$ , if ${\mathrm {sTD}}$ holds for $\mathbf {\Sigma }^1_n$ -sets, then every $\mathbf {\Sigma }^1_n$ -set is measurable.

5.2 The proof of part (2)

We first prove the following lemma.

Lemma 5.2. Assume ${\mathrm {ZF}}+{\mathrm {sTD}}$ . For every perfect set P of reals and every partition $P=\bigcup _{n<\omega } B_n$ , there exists n such that $B_n$ has a perfect subset.

Proof Clearly we may assume that $P=2^\omega $ via a homeomorphism. Then for some n, $B_n$ ranges Turing degrees cofinally. By sTD, $B_n$ contains a perfect subset.

Proof of part (2) of Theorem 5.1

Suppose that A is uncountable. By Lemma 3.3, we may fix a perfect set B and a sequence of functions $\{f_n\}_{n\in \omega }$ as in the lemma. Then by Lemma 5.2, we can choose a perfect $Q\subset f^{-1}_n[A]$ for some n. Now $f_n[Q]$ is an uncountable analytic subset of A. So $f_n[Q]$ and hence A contains a perfect subset.

Remark. We would like to thank the referee who pointed out to us that the proof derived the result that ${\mathrm {ZF}}+{\mathrm {TD}}+\mbox {"every set of reals has Baire property"}$ implies every uncountable set of reals has a perfect subset.

Here we mention another approach, within ${\mathrm {ZF}}+{\mathrm {sTD}}+{\mathrm {DC}_{\mathbb {R}}}$ , to get a perfect subset due to Sami. A set A of reals is called Bernstein if neither A nor $\mathbb {R}\setminus A$ has a perfect subset. Notice that the nonexistence of a Bernstein set implies that for every perfect set P and its subset $A\subseteq P$ , either A or $P\setminus A$ has a perfect subset. Sami observed the following relationship between the existence of a Bernstein set and perfect subset property.

Lemma 5.3 (Sami [23])

Assume ${\mathrm {ZF}}+{\mathrm {TD}}+{\mathrm {DC}_{\mathbb {R}}}$ . If there is no Bernstein set, then every uncountable set of reals has a perfect subset.

Proof Suppose that A is uncountable. By Lemma 3.3, we may fix a perfect set B and a sequence of functions $\{f_n\}_{n\in \omega }$ as in the lemma.

Let $T^0\subseteq 2^{<\omega }$ be a perfect tree such that $[T^0]=B$ . We will inductively choose a decreasing sequence of perfect trees $T^0\supset T^1\supset \cdots \supset T^n\supset \cdots $ until the procedure terminates. Suppose that $T^n$ has been chosen and we choose $T^{n+1}$ .

Case $(1)$ . There is some perfect tree $T^{*}\subseteq T^n$ such that $f_n$ is defined on every member in $[T^*]$ and $f_n([T^*])\subseteq A$ . Fix such $T^*$ . Then $f_n([T^*])\subseteq A$ is an uncountable analytic set. Thus A must have a perfect subset. The procedure terminates and we are done.

Case $(2)$ . Otherwise. Then by the assumption of no Bernstein set, choose $T^{n+1}$ to be a perfect subtree of $ T^n$ such that for every $x\in [T^{n+1}]$ , either $f_n(x)$ is not defined or $f_n(x)\not \in A$ .

Either we stop at Case (1) of some n, then we find a perfect subset of A. Or else, the construction goes through all of n’s. Then $\bigcap _{n<\omega } [T^n]$ is non-empty. Moreover, for every $x\in \bigcap _{n<\omega } [T^n]$ and every n, either $f_n(x)$ is not defined or $f_n(x)\not \in A$ . This contradicts the fact that $B\subseteq \bigcup _{n\in \omega } f^{-1}_n(A)$ .

Thus we must stop at Case (1) of some n and so A must have a perfect subset.

${\mathrm {sTD}}$ implies that every set is measurable and so there is no Bernstein set. Thus ${\mathrm {ZF}}+{\mathrm {sTD}}+{\mathrm {DC}_{\mathbb {R}}}$ implies every uncountable set of reals has a perfect subset.

5.3 An application of regularity properties to linear orderings over $\mathbb {R}$

Lemma 5.4. Assume ${\mathrm {ZF}}+{\mathrm {CC}_{\mathbb {R}}}+\mbox {"every set of reals is measurable."}$ Given any linear order $\leq _L$ of $\mathbb {R}$ and any non-null set $A\subseteq \mathbb {R}$ , the collection of all $x\in A$ such that either $\{y\in A\mid y\leq _L x\}$ is null or $\{y\in A\mid x\leq _L y\}$ is null is a null set.

Proof Given a linear order $\leq _L$ over $\mathbb {R}$ , let $A\subseteq \mathbb {R}$ be any non-null set. Fix a non-null set $B\subseteq A$ . By Fubini’s theorem, the set $\{(x,y)\mid x\leq _L y\wedge x\in B\wedge y\in B\}$ is measurable and has positive measure. Let

$$ \begin{align*}L^B=\{x\in B\mid \{y\in A\mid y\leq_L x\} \mbox{ is null}\}\end{align*} $$

be a subset of B. Then by Fubini’s theorem again, the set $B\setminus L^B$ is not null. So the set

$$ \begin{align*}L^A=\{x\in A\mid \{y\in A\mid y\leq_L x\} \mbox{ is null}\}\end{align*} $$

is a null subset of A.

By the same method, the set

$$ \begin{align*}R^A=\{x\in A\mid \{y\in A\mid x\leq_L y\} \mbox{ is null}\}\end{align*} $$

is also a null subset of A.

Finally, we have the following basis theorem for linear orderings over $\mathbb {R}$ under ${\mathrm {ZF}}+{\mathrm {sTD}}$ . In what follows, we use $\leq $ to denote the usual order on reals. Since the lexicographic order on $2^\omega $ is order isomorphic to the real order on the Cantor set through the standard bijection, we also use $\leq $ to denote this order on $2^\omega $ .

Theorem 5.5. Assume ${\mathrm {ZF}}+{\mathrm {DC}_{\mathbb {R}}}+\mbox {"every set of reals is measurable."}$ For every linear order $\leq _L$ over $\mathbb {R}$ , there is an order-preserving embedding from $(2^{\omega },\leq )$ to $(\mathbb {R},\leq _L)$ .

Proof First we set $P_{\emptyset }=[0, 1]$ .

By Lemma 5.4, there is a real $x\in P_{\emptyset }$ such that both the sets $\{y\in A\mid y\leq _L x\}$ and $\{y\in A\mid x\leq _L y\}$ have positive measure. So both of them have disjoint perfect subsets $P_0$ and $P_1$ with positive measure, respectively. Moreover, we may require that for any $i\in \{0,1\}$ and $y,z\in P_{i}$ , $|y-z|\leq 2^{-1}$ .

Now, by an induction, it is not difficult to construct a sequence $\{P_{\sigma }\}_{\sigma \in 2^{<\omega }}$ of perfect sets so that:

• • If $\sigma $ extends $\tau $ , written as $\sigma \succ \tau $ , then $P_{\sigma }\subset P_{\tau }$ has positive measure.

• • If $\sigma $ and $\tau $ are incompatible, then $P_{\sigma }\cap P_{\tau }=\emptyset $ .

• • If for some n, $\sigma \upharpoonright _n=\tau \upharpoonright _n$ and $\sigma (n)<\tau (n)$ , then $\forall x\in P_{\sigma }\forall y\in P_{\tau }(x\leq _L y)$ .

• • For any $\sigma $ and $x,y\in P_{\sigma }$ , $|x-y|\leq 2^{-|\sigma |}$ .

Define $f:2^{\omega }\to \mathbb {R}$ so that $f(x)$ is the unique real in $\bigcap _n P_{x{\upharpoonright} n}$ . Then f is an order-preserving embedding from $(2^{\omega },\leq )$ to $(\mathbb {R},\leq _L)$ .

One may wonder what happens to Lemma 5.4 under ${\mathrm {ZF}}+{\mathrm {TD}}$ . Since it is unknown whether ${\mathrm {ZF}}+{\mathrm {TD}}$ implies that every set of reals is measurable, we have to use a more involved argument.

Definition 5.6. A linear order $(L,\leq _L)$ is locally countable if for any $l\in L$ , the set $\{x\leq _L l\mid x\in L\}$ is countable.

A typical locally countable order is $(\omega _1,\leq )$ .

For a set A of reals that is closed under Turing equivalence relation, a real x is a minimal upper bound of A if:

• • every member of A is recursive in x; and

• • there is no real $y<_T x$ such that every member of A is recursive in y.

By a classical theorem in recursion theory (see Theorem 4.11 in [14]), every countable set of reals has a minimal upper bound.

Lemma 5.7. Assume ${\mathrm {ZF}}+{\mathrm {TD}}$ . There is no uncountable set $A\subseteq \mathbb {R}$ with a locally countable linear order over A.

Proof By Proposition 3.2, it suffices to prove that there is no locally countable linear order on $\mathbb {R}$ .

Suppose not. Let $(\mathbb {R},\leq _L)$ be a locally countable order. For every real x, let $I_x$ be the Turing downward closure of the set $\{z\mid z\leq _L x\}$ . I.e.,

$$ \begin{align*}I_x=\{s\mid \exists z\leq_L x(s\leq_T z)\}.\end{align*} $$

Obviously $x\leq _L y$ implies $I_x\subseteq I_y$ .

Note that for any real z, there is a real x such that $z\in I_x$ . So there is a real $z_0\geq _T z$ such that $z_0$ is a minimal upper bound of $I_x$ . By ${\mathrm {TD}}$ , there is a real $z_1$ such that every real $z_2\geq _T z_1$ is a minimal upper bound over $I_x$ for some x.

For every real z, let

$$ \begin{align*}M_z=\{x\mid z\mbox{ is a minimal upper bound of }I_x)\}\end{align*} $$

and

$$ \begin{align*}N_z=\bigcup_{x\in M_z}I_x.\end{align*} $$

Note that $M_{z_2}$ is nonempty for every $z_2\geq _T z_1$ . We have the following fact:

• • For any $z_2, z_3\geq _T z_1$ , either $N_{z_3}\subseteq N_{z_2}$ or $N_{z_2}\subseteq N_{z_3}$ . [To see this, suppose that $N_{z_3}\not \subseteq N_{z_2}$ . Then there must be some $x_3\in M_{z_3}$ such that for any $x_2\in M_{z_2}$ , $x_3\not \leq _L x_2$ . In other words, $x_2\leq _L x_3$ for any $x_2\in M_{z_2}$ . So $N_{z_2}\subseteq N_{z_3}$ .]

Now fix a pair of minimal covers $z_2\not \equiv _T z_3$ of $z_1$ (i.e., for $i\in \{2,3\}$ , $z_i>_T z_1$ but there is no real y strictly between $z_1$ and $z_i$ in the Turing reduction order sense. For the existence of such a pair, see Lemma 2.1). By the fact above, WLOG, we may assume $N_{z_2}\subseteq N_{z_3}$ and fix some $x\in M_{z_2}$ . Then every real in $I_x\subseteq N_{z_2}\subseteq N_{z_3}$ is recursive in both $z_2$ and $z_3$ . So every real in $I_x$ is recursive in $z_1$ . This contradicts the fact that $z_2$ is a minimal upper bound of $I_x$ and $z_1<_T z_2$ .

Corollary 5.8. Assume ${\mathrm {ZF}}+{\mathrm {TD}}$ . For every uncountable set $A\subseteq \mathbb {R}$ and linear order $\leq _L$ over A, there are uncountably many reals $x\in A$ such that both $\{y\in A\mid y\leq _L x\}$ and $\{y\in A\mid x\leq _L y\}$ are uncountable.

Proof Given a linear order $\leq _L$ over $\mathbb {R}$ , let

$$ \begin{align*}L=\{x\in A\mid \{y\in A\mid y\leq_L x\} \mbox{ is countable}\}\end{align*} $$

and

$$ \begin{align*}R=\{x\in A\mid \{y\in A\mid x\leq_L y\} \mbox{ is countable}\}.\end{align*} $$

By Lemma 5.7, both L and R are countable. So there are uncountably many reals $x\in A$ such that both $\{y\mid y\leq _L x\}$ and $\{y\mid x\leq _L y\}$ are uncountable.

Now we may obtain the following result.

Theorem 5.9. Assume ${\mathrm {ZF}}+{\mathrm {TD}}+{\mathrm {DC}_{\mathbb {R}}}$ . The following are equivalent.

1. 1. Every uncountable set of reals has a perfect subset.

2. 2. For every linear order $\leq _L$ over $\mathbb {R}$ , there is an order-preserving embedding from $(2^{\omega },\leq )$ to $(\mathbb {R},\leq _L)$ .

Proof (1) $\Rightarrow $ (2). The argument of Theorem 5.5 works here. Just replace “set with positive measure” by “uncountable set.”

(2) $\Rightarrow $ (1). Fix an uncountable set of reals A. By Proposition 3.2, $|A|=|\mathbb {R}|$ . So $(A, \leq )$ is order isomorphic to $(\mathbb {R}, \leq _L)$ for some $\leq _L$ . By (2), there is an order-preserving map from $(2^\omega , \leq )$ to $(\mathbb {R}, \leq _L)$ and hence $(A, \leq )$ .

Fix $\pi : 2^\omega \to A$ that preserves order and so is monotonic. Then $\pi $ is continuous on all but countably many points. In particular, $\pi $ is continuous (and injective) on a perfect subset P. So $\pi [P]$ is a perfect subset of A.

6 Regularity properties for dimension theory

For the notions and terminologies in fractal geometry, we follow the book [7].

Given a non-empty $U\subseteq \mathbb {R}$ , the diameter of U is

$$ \begin{align*}diam(U)=|U|=\sup\{|x-y| : x, y\in U\}.\end{align*} $$

Given any set $E\subseteq \mathbb {R}$ and $d\geq 0$ , let

$$ \begin{align*}\mathcal{H}^d(E)=\lim_{\delta\to 0}\inf\{\sum_{i<\omega}|U_i|^{d}: \{U_i\}\mbox{ is an open cover of } E \wedge \forall i\ |U_i|<\delta\},\end{align*} $$

$$ \begin{align*} \mathcal{P}_0^d(E)=\lim_{\delta\to 0}\sup\{\sum_{i<\omega} |B_i|^d: \{B_i\}\mbox{ is a collection of disjoint balls of radii at}\\ \mbox{ most }\delta \mbox{ with centres in }E\}, \end{align*} $$

and

$$ \begin{align*}\mathcal{P}^d(E)=\inf\{\sum_{i<\omega}\mathcal{P}^d_0(E_i)\mid E\subseteq \bigcup_{i<\omega} E_i\}.\end{align*} $$

Definition 6.1. Given any set $E$ :

1. (1) The Hausdorff dimension of E, or $\mathrm {Dim_H}(E)$ , is

   $$ \begin{align*}\inf\{d\mid \mathcal{H}^d(E)=0\}.\end{align*} $$

2. (2) The Packing dimension of E, or $\mathrm {Dim_P}(E)$ , is

   $$ \begin{align*}\inf\{d\mid \mathcal{P}^d(E)=0\}.\end{align*} $$

By the same reason as in the Lebesgue measure, it can be proved with ${\mathrm {ZF}}+{\mathrm {CC}_{\mathbb {R}}}$ that for every Borel set B and $\epsilon>0$ , there is a closed set $F\subseteq B$ such that $\mathrm {Dim_H}(F)>\mathrm {Dim_H}(B)-\epsilon $ .

Theorem 6.2 (Besicovitch [1] and Davis [5])

For every analytic set A and every $\epsilon>0$ , there is a closed set $F\subseteq A$ such that $\mathrm {Dim_H}(F)\geq \mathrm {Dim_H}(A)- \epsilon $ .

Theorem 6.3 (Joyce and Preiss [11])

For every analytic set A and every $\epsilon>0$ , there is a closed set $F\subseteq A$ such that $\mathrm {Dim_P}(F)\geq \mathrm {Dim_P}(A)- \epsilon $ .

However Slaman proves that both Theorems 6.2 and 6.3 may fail even for some $\Pi ^1_1$ set under certain assumptions.

Theorem 6.4 (Slaman [25])

Suppose that the set of constructible reals is not null, then there is a $\Pi ^1_1$ set C with $\mathrm {Dim_H}(C)=1$ but for any Borel $F\subset C$ , $\mathrm {Dim_P}(F)=0$ .

We prove that Theorems 6.2 and 6.3 both remain true for all sets of reals under ${\mathrm {ZF}}+{\mathrm {sTD}}$ . As pointed out by one of the referees, the following result can be localized similarly as the localization of Theorem 5.1.

Theorem 6.5. ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that for every set of reals A and every $\epsilon>0:$

1. (1) There is a closed set $F\subseteq A$ such that $\mathrm {Dim_H}(F)\geq \mathrm {Dim_H}(A)-\epsilon $ .

2. (2) There is a closed set $F\subseteq A$ such that $\mathrm {Dim_P}(F)\geq \mathrm {Dim_P}(A)-\epsilon $ .

To show the theorem, we use the “point-to-set” method.

Some more facts from algorithmic randomness theory are needed. Let K denote the prefix-free Kolmogorov complexity. We use $K^x$ to denote the prefix-free Kolmogorov complexity with oracle, which is a real, x. The following “point to set” style theorem is due to Lutz and Lutz. A similar form was also discovered by Cutler. See [4, Theorem 1.4].

Theorem 6.6 (Lutz and Lutz [15])

For every set $A\subseteq \mathbb {R}$ ,

$$ \begin{align*}\mathrm{Dim_H}(A)=\inf_{x\in\mathbb{R}} \sup_{y\in A} \underline\lim_{n\to \infty}\frac{K^x(y{\upharpoonright} n)}{n}\end{align*} $$

and

$$ \begin{align*}\mathrm{Dim_P}(A)=\inf_{x\in\mathbb{R}} \sup_{y\in A} \overline\lim_{n\to \infty}\frac{K^x(y{\upharpoonright} n)}{n}.\end{align*} $$

The following lowness property is crucial to our proof.

Theorem 6.7 (Herbert [9]; Lempp, Miller, Ng, Turetsky, Weber [13])

• • There is a perfect tree $T\subseteq 2^{<\omega }$ recursive in $\emptyset '$ such that for any real $x\in [T]$ ,

  $$ \begin{align*}\forall y\in \mathbb{R}\Bigg(\underline{\lim}_{n\to \infty}\frac{K(y{\upharpoonright} n)}{n}=\underline{\lim}_{n\to \infty}\frac{K^x(y{\upharpoonright} n)}{n}\Bigg).\end{align*} $$

• • There is a perfect tree $T\subseteq 2^{<\omega }$ recursive in $\emptyset '$ such that for any real $x\in [T]$ ,

  $$ \begin{align*}\forall y\in \mathbb{R}\Bigg(\overline{\lim}_{n\to \infty}\frac{K(y{\upharpoonright} n)}{n}=\overline{\lim}_{n\to \infty}\frac{K^x(y{\upharpoonright} n)}{n}\Bigg).\end{align*} $$

Now we are ready to prove our major theorem of this section.

Proof of Theorem 6.5

(1). Suppose that $A\subseteq \mathbb {R}$ with $\mathrm {Dim_H}(A)>0$ . Fix any $\epsilon>0$ . By Theorem 6.6, for every real z, there is some real $x\in A$ such that

$$ \begin{align*}\underline{\lim}_{n\to \infty}\frac{K^z(x{\upharpoonright} n)}{n}>\mathrm{Dim_H}(A)-\frac{\epsilon}{2}.\end{align*} $$

By Theorem 6.7 relative to z, there is a real $y>_T z$ such that

$$ \begin{align*}\underline{\lim}_{n\to \infty}\frac{K^y(x{\upharpoonright} n)}{n}=\underline{\lim}_{n\to \infty}\frac{K^z(x{\upharpoonright} n)}{n}>\mathrm{Dim_H}(A)-\frac{\epsilon}{2} \wedge y'>_T x.\end{align*} $$

So there must be some $e_0$ such that the set

$$ \begin{align*}B_{e_0}=\Bigg\{y\mid \Phi_{e_0}^{y'}\in A\wedge \underline{\lim}_{n\to \infty}\frac{K^y(\Phi_{e_0}^{y'}{\upharpoonright} n)}{n}>\mathrm{Dim_H}(A)-\frac{\epsilon}{2}\Bigg\}\end{align*} $$

ranges Turing degrees cofinally. By ${\mathrm {sTD}}$ , there is a pointed set $P\subseteq B_{e_0}$ .

Then the set

$$ \begin{align*}C=\{x\mid \exists y\in P(\Phi_{e_0}^{y'}=x)\}\end{align*} $$

is an analytic subset of A. By Theorem 6.6,

$$ \begin{align*}\mathrm{Dim_H}(C)>\mathrm{Dim_H}(A)-\frac{\epsilon}{2}.\end{align*} $$

By Theorem 6.2, C has a closed subset F such that

$$ \begin{align*}\mathrm{Dim_H}(F)>\mathrm{Dim_H}(C)-\frac{\epsilon}{2}.\end{align*} $$

Thus

$$ \begin{align*}\mathrm{Dim_H}(F)>\mathrm{Dim_H}(A)-\epsilon.\end{align*} $$

(2). The same proof as (1) except replacing Hausdorff dimension with packing dimension. We leave the details to readers.

To continue our study, we need the following folklore technique lemma of which we sketch a proof for the completeness.

Lemma 6.8 (Folklore)

Assume ${\mathrm {ZF}}+{\mathrm {sTD}}$ . If $f:\mathbb {R}\to Ord$ is a degree invariant $($ i.e., $x\equiv _T y\implies f(x)=f(y))$ map such that $f(x)<\omega _1^x$ , then there is an ordinal $\alpha $ such that $f(x)=\alpha $ over an upper cone of Turing degrees.

Proof Fix such a map f. Since there are countably many recursive functionals, by ${\mathrm {sTD}}$ , there is some recursive functional $\Phi $ such that $\Phi ^x$ codes a linear order for every real x, and a pointed set P such that $f(x)\cong \Phi ^x$ for any $x\in P$ . Let T be a tree representing P such that $\forall x\in P(T\leq _T x)$ . Then the set

$$ \begin{align*}\{\Phi^x\mid x\in P\}\end{align*} $$

is a $\Sigma ^1_1(T)$ set and so $\Phi ^x$ represents an ordinal smaller than $\omega _1^T$ for any $x\in P$ by $\Sigma ^1_1$ -boundedness relative to T (see [2]). By ${\mathrm {sTD}}$ again, there must be some $\alpha <\omega _1^T$ and a pointed set $Q\subseteq P$ such that $f(x)=\alpha $ for any $x\in Q$ . This finishes the proof.

Crone, Fishman, and Jackson also proved the following result.

Theorem 6.9 (Crone, Fishman, and Jackson [3])

Assume ${\mathrm {ZF}}+{\mathrm {AD}}+\mathrm {DC}$ . If $A=\bigcup _{\alpha <\kappa }A_{\alpha }$ for some ordinal $\kappa $ , then $\mathrm {Dim_H}(A)=\sup \{\mathrm {Dim_H}( A_{\alpha })\mid \alpha <\kappa \}$ .

We may provide an “elementary” proof of the following weaker result under ${\mathrm {ZF}}+{\mathrm {sTD}}$ .

Theorem 6.10. Assume ${\mathrm {ZF}}+{\mathrm {sTD}}$ . If $A=\bigcup _{\alpha <\omega _1}A_{\alpha }$ , then

$$ \begin{align*}\mathrm{Dim_H}(A)=\sup\{\mathrm{Dim_H}( A_{\alpha})\mid \alpha<\omega_1\} \mbox{ and }\mathrm{Dim_P}(A)=\sup\{\mathrm{Dim_P}( A_{\alpha})\mid \alpha<\omega_1\}.\end{align*} $$

Proof Let $r=\mathrm {Dim_H}(A)$ and for every real x,

$$ \begin{align*}\gamma_x=\min\Bigg\{\gamma|\sup_{y\in \bigcup_{\alpha< \gamma}A_{\alpha}} \underline\lim_{n\to \infty}\frac{K^x(y{\upharpoonright} n)}{n}\geq r\Bigg\}.\end{align*} $$

By Theorem 6.6, $\gamma _x$ is defined for every real x.

For any real z, by Theorem 6.7 and the assumption, there is a real $x>_T z$ such that $\gamma _x=\gamma _z$ but $\omega _1^{x'}>\gamma _z$ . So

$$ \begin{align*}\gamma_x=\gamma_z<\omega_1^{x'}=\omega_1^{x}.\end{align*} $$

In other words, $x\mapsto \gamma _x$ is a degree invariant function such that $\gamma _x<\omega _1^x$ over an upper cone of Turing degrees. Then by Lemma 6.8, $x\mapsto \gamma _x$ is a constant, say $\eta $ , over an upper cone. Then, by the countability of $\eta $ , for every $m\in \omega $ , there must be some $\alpha _m<\eta $ such that the set $\{x\mid \sup _{y\in A_{\alpha _m}} \underline \lim _{n\to \infty }\frac {K^x(y{\upharpoonright} n)}{n}\geq r-\frac {1}{m}\}$ ranges Turing degrees cofinally. Then by Theorem 6.6, $\mathrm {Dim_H}(A_{\alpha _m})\geq r-\frac {1}{m}$ . So

$$ \begin{align*}\mathrm{Dim_H}(A)=\sup\{\mathrm{Dim_H}( A_{\alpha})\mid \alpha<\eta\}=\sup\{\mathrm{Dim_H}( A_{\alpha})\mid \alpha<\omega_1\}.\end{align*} $$

We leave the proof of the second part to readers.

Some remarks:

• • The conclusion of Theorem 6.10 can also be proved within $\mathrm {ZFC}+\mathrm {MA}_{\aleph _1}$ . Furthermore, $\omega _1$ can be replaced by any cardinal $\kappa <2^{\aleph _0}$ .

• • We don’t know the consistency strength of the conclusions of Theorem 6.5.