Nearly computable real numbers

Abstract

We call a sequence ${(a_{n})}_{n}$ of elements of a metric space nearly computably Cauchy if for every increasing computable function $r : N \to N$ the sequence ${(d (a_{r (n + 1)}, a_{r (n)}))}_{n}$ converges computably to 0. We show that there exists an increasing sequence of rational numbers that is nearly computably Cauchy and unbounded. Then we call a real number α nearly computable if there exists a computable sequence ${(a_{n})}_{n}$ of rational numbers that converges to α and is nearly computably Cauchy. It is clear that every computable real number is nearly computable, and it follows from a result by Downey and LaForte (Theoretical Computer Science 284 (2002) 539–555) that there exists a nearly computable and left-computable number that is not computable. We observe that the set of nearly computable real numbers is a real closed field and closed under computable real functions with open domain, but not closed under arbitrary computable real functions. Among other things we strengthen results by Hoyrup (Theory of Computing Systems 60 (2017) 396–420) and by Stephan and Wu (In New computational paradigms. First conference on computability in Europe, CiE 2005, Proceedings (2005) 461–469 Springer) by showing that any nearly computable real number that is not computable is weakly 1-generic (and, therefore, hyperimmune and not Martin-Löf random) and strongly Kurtz random (and, therefore, not K-trivial), and we strengthen a result by Downey and LaForte (Theoretical Computer Science 284 (2002) 539–555) by showing that no promptly simple set can be Turing reducible to a nearly computable real number.

Keywords

Nearly computable numbers computable convergence real closed field weakly 1-generic promptly simple

1. Introduction

A real number x is called computable if there exists a computable sequence ${(q_{n})}_{n}$ of rational numbers with $| x - q_{n} | ⩽ 2^{- n}$ , for all n. From a computability-theoretic point of view this set is certainly the most important subset of the field of real numbers. But there are also larger subsets that are of interest from a computability-theoretic point of view. For example, a real number x is called computably approximable if there exists a computable sequence of rational numbers that converges to x. And a real number x is called left-computable if there exists an increasing computable sequence of rational numbers that converges to x. Let $EC$ denote the set of all computable numbers, $WC$ denote the set of all differences of two left-computable numbers, and $CA$ denote the set of all computably approximable numbers. It is well known that $EC ⊊ WC ⊊ CA$ (see [2]), and that all three subsets $EC$ , $WC$ , and $CA$ are real closed subfields of the field $R$ of all real numbers (see [25, Theorem 6.3.10] and [18]). The set $LC$ of left-computable numbers plays an important role in the theory of algorithmic randomness; see [4,17]. And in a series of papers Rettinger and Zheng and other coauthors have analyzed various computability-theoretic sets of real numbers, mostly between $WC$ and $CA$ ; for an overview see [19].

In this article we consider a new computability-theoretic set of real numbers that is only slightly larger than the set $EC$ of computable real numbers. We call a sequence ${(a_{n})}_{n}$ of elements of a metric space nearly computably Cauchy if it has the property that for every increasing computable function $r : N \to N$ the sequence ${(d (a_{r (n + 1)}, a_{r (n)}))}_{n}$ converges computably to 0. We call a real number α nearly computable if there exists a computable sequence ${(a_{n})}_{n}$ of rational numbers that converges to α and is nearly computably Cauchy. The idea behind this is the following. The limit of any convergent sequence ${(a_{n})}_{n}$ of real numbers can be written as the limit of the series $\sum_{i = 0}^{\infty} b_{i}$ , where $b_{i} : = a_{i} - a_{i - 1}$ , for $i > 0$ , and $b_{0} : = a_{0}$ . Then the sequence ${(b_{n})}_{n}$ converges to zero. We are interested in the computability-theoretic properties of the real numbers that one obtains as the limit of such a series when one imposes conditions on the convergence of the sequence ${(b_{n})}_{n}$ . In this article we consider a strong condition by demanding that for any increasing function $r : N \to N$ the sequence ${(\sum_{i = r (n) + 1}^{r (n + 1)} b_{i})}_{n}$ converges to zero computably.

In the following paragraph we give an overview of the content of the paper. The main results are highlighted by bullets. In the two short Sections 2 and 3 we first introduce some notation and then some basic notions concerning sequences in metric spaces. Then we define computable sequences and list some properties of computable sequences. In Section 5 we give various characterizations of nearly computably Cauchy sequences in a metric space and in a normed vector space. At the conference CCA 2018, after the presentation of [12], M. Schröder asked the following question: is every nearly computably Cauchy sequence necessarily a Cauchy sequence? On the one hand, we show that this is true for nearly computably Cauchy sequences that are additionally computable. On the other hand, we show that

there exists an increasing sequence of rational numbers that is nearly computably Cauchy and unbounded and, therefore, not a Cauchy sequence (Theorem 5.6).

We also make some observations about nearly computably Cauchy sequences that will turn out to be useful later. In Section 6 we study the set of all nearly computable real numbers. It is a proper superset of the set of all computable real numbers. We show that

the set of all nearly computable real numbers is a real closed field (Corollary 6.5).

In fact, it is closed under computable functions with open domain. (Proposition 6.5).

But we also show that it is not closed under arbitrary computable functions (Theorem 6.8).

In Section 7 we give several characterizations of real numbers that are left-computable and nearly computable. We note that it follows from a result by Downey and LaForte [7] that there exists a nearly computable and left-computable number that is not computable. We also discuss two cases where the requirement that a real number should be nearly computable and left-computable and satisfy some additional condition forces the real number to be even computable. For example, an argument by Downey, Hirschfeldt, and LaForte [5, Pages 105, 106] shows that any strongly left-computable and nearly computable real number is even computable. In Section 8 we discuss three further results of the same type, one by Hoyrup [11] and two by Stephan and Wu [24]. Hoyrup [11] has shown that any nearly computable and left-computable number that is not computable is weakly 1-generic. We show that

in this result the assumption that the real number should be left-computable can be omitted (Theorem 8.3).

As any weakly 1-generic real is hyperimmune [14] and not Martin-Löf random we obtain a similar strengthening of a result by Stephan and Wu [24], who had shown that any nearly computable number that is left computable is hyperimmune (and, therefore, not Martin-Löf random). Finally, also

in their result that any nearly computable number that is left computable is strongly Kurtz random (and, therefore, not K-trivial) one can omit the assumption that the real number should be left-computable (Theorem 8.8).

In the final section, Section 9, we strengthen an observation by Downey and LaForte [7] in a similar way by showing that

no promptly simple set can be Turing reducible to a nearly computable real number (Theorem 9.3).

Some of the results of this paper have been presented at the conference CCA 2018 [12] (Corollary 6.7 and Theorem 7.2, together with other results not contained in this paper), and at the conference CCA 2022 [9] (Proposition 5.5 and Theorem 5.6, Theorem 6.8, Corollary 8.4 and Theorem 8.8, Theorem 9.3). Note that each of the extended abstracts [12] and [9] is just two pages long and does not contain any proofs.

2. Preliminaries

Let X and Y be sets. A subset $f \subseteq X \times Y$ with the property that for any $x \in X$ there is at most one $y \in Y$ with $(x, y) \in f$ is called a function $f : \subseteq X \to Y$ with domain $dom (f) : = {x \in X : (\exists y \in Y) (x, y) \in f}$ . Instead of $(x, y) \in f$ we write $f (x) = y$ . For $x \in X$ we write $f (x) ↓$ if $x \in dom (f)$ and $f (x) ↑$ if $x \notin dom (f)$ . If $dom (f) = X$ then we may write the function f as $f : X \to Y$ and call it a total function. Let $Y^{X}$ be the set of all total functions from X to Y. Note that in the case $X = \emptyset$ there is exactly one function in $Y^{X}$ , the empty function. For $k \in N$ let $X^{k} : = X^{{0, \dots, k - 1}}$ be the set of all sequences of length k of elements of X. For $k = 0$ this set contains exactly one element, the empty sequence. Let $X^{*} : = ⋃_{k = 0}^{\infty} X^{k}$ be the set of all finite sequences of elements of X. Let $N = {0, 1, 2, \dots}$ as usual be the set of nonnegative integers or natural numbers. The elements of $X^{N}$ are called sequences with elements in X, and a sequence $p : N \to X$ is often written in the form ${(p (n))}_{n}$ or ${(p_{n})}_{n}$ . A sequence ${(x_{n})}_{n}$ of real numbers is called nondecreasing if $x_{n} ⩽ x_{n + 1}$ , for all n, and it is called increasing if $x_{n} < x_{n + 1}$ for all n. For any $k \in N$ and $p \in X^{N}$ let $p ↾ k$ be the prefix of length k of the sequence $p : N \to X$ , that is, the uniquely determined element $g \in X^{k}$ with $g \subseteq p$ , in other words, $p ↾ k = p (0) \dots p (k - 1) = p_{0} \dots p_{k - 1}$ . Sometimes we will identify a subset $A \subseteq N$ with its characteristic function $χ_{A} : N \to {0, 1}$ defined by $\begin{array}{r} χ_{A} (n) : = \{\begin{array}{ll} 1 & if n \in A, \\ 0 & if n \notin A, \end{array} \end{array}$ for $n \in N$ . The function $⟨ \cdot, \cdot ⟩ : N^{2} \to N$ defined by $⟨ i, j ⟩ : = \frac{(i + j) \cdot (i + j + 1)}{2} + i$ , for all $i, j \in N$ , is a computable bijection between $N^{2}$ and $N$ . Also its projections $π_{1} : N \to N$ and $π_{2} : N \to N$ defined by $π_{1} (⟨ i, j ⟩) : = i$ and $π_{2} (⟨ i, j ⟩) : = j$ , for all $i, j \in N$ , are computable. We can extend this to a bijection $⟨ \dots ⟩ : N^{k} \to N$ for any $k ⩾ 1$ by defining $⟨ i ⟩ : = i$ , for any i, and by $⟨ m_{1}, \dots, m_{k}, m_{k + 1} ⟩ : = ⟨ ⟨ m_{1}, \dots, m_{k} ⟩, m_{k + 1} ⟩$ , for $k ⩾ 2$ and $m_{1}, \dots, m_{k}, m_{k + 1} \in N$ .

Let $φ_{0}, φ_{1}, φ_{2}, \dots$ be a standard enumeration of all possibly partial computable functions with domain and range in $N$ . For every $e \in N$ we write $W_{e} : = dom (φ_{e})$ . Note that then $W_{0}, W_{1}, W_{2}, \dots$ is an effective list of all c.e. subsets of $N$ . Let $h_{u} : N \to N^{2}$ be an injective computable function with $h_{u} (N) = {(e, n) \in N^{2} | φ_{e} (n) ↓}$ . For every $s \in N$ we write $W_{e} [s] : = {n \in N : (e, n) \in {h_{u} (0), \dots, h_{u} (s - 1)}}$ . Furthermore, sometimes for a c.e. set D and for $s \in N$ we use the expression $D [s]$ . Then it is understood that we have fixed some e with $D = W_{e}$ and write $D [s]$ for $W_{e} [s]$ .

Let $Q$ be the set of all rational numbers and $R$ be the set of all real numbers. In order to transfer computability notions from $N$ to $Q$ we use the following surjective function $ν_{Q} : N \to Q$ defined by $\begin{array}{r} ν_{Q} (⟨ a, b ⟩) : = \{\begin{array}{ll} \frac{- a / 2}{b + 1} & if a is even, \\ \frac{(a + 1) / 2}{b + 1} & if a is odd, \end{array} \end{array}$ for $a, b \in N$ . Instead of $ν_{Q}$ one may use any equivalent numbering of $Q$ ; see the end of Section 4. For introducing useful computability notions on $R$ even any equivalent numbering of a dense subset of $R$ would work as well; see again the end of Section 4. For any $k ⩾ 1$ we define a surjective function $ν_{Q}^{(k)} : N \to Q^{k}$ by $ν_{Q}^{(k)} (⟨ i_{1}, \dots, i_{k} ⟩) : = (ν_{Q} (i_{1}), \dots, ν_{Q} (i_{k}))$ , for $i_{1}, \dots, i_{k} \in N$ (note that $ν_{Q}^{(1)} = ν_{Q}$ ).

3. Computably convergent sequences and computably Cauchy sequences

A metric space is a pair $(X, d)$ consisting of a nonempty set X and a function $d : X \times X \to R$ , the metric or distance function, satisfying the following three conditions:

$(\forall x, y \in X) d (x, y) = 0 ⟺ x = y$ ,

$(\forall x, y \in X) d (x, y) = d (y, x)$ (symmetry),

$(\forall x, y, z \in X) d (x, z) ⩽ d (x, y) + d (y, z)$ (triangle inequality).

We note that, if $(X, d)$ is a metric space then, for all $x, y \in X$ , we have $d (x, y) ⩾ 0$ (indeed, $d (x, y) = \frac{1}{2} \cdot (d (x, y) + d (y, x)) ⩾ \frac{1}{2} \cdot d (x, x) = 0$ ). We remind the reader of some classical notions and observations for sequences in X.

Definition 3.1.
Let $(X, d)$ be a metric space. A sequence ${(x_{n})}_{n}$ in X converges to an element $y \in X$ if there exists a total function $g : N \to N$ such that, for all $m, n \in N$ , $\begin{array}{r} if m ⩾ g (n) then d (x_{m}, y) ⩽ 2^{- n} . \end{array}$ A function g with this property is called a modulus of convergence of the sequence ${(x_{n})}_{n}$ .

It is easy to see and well known that, if a sequence ${(x_{n})}_{n}$ in a metric space $(X, d)$ converges to an element $y \in X$ and to an element $z \in X$ then $y = z$ . Thus, if a sequence ${(x_{n})}_{n}$ in X converges to an element y then this element is uniquely determined. We call it the limit of ${(x_{n})}_{n}$ and write it ${lim}_{n \to \infty} x_{n}$ .
Definition 3.2.
Let $(X, d)$ be a metric space. A sequence ${(x_{n})}_{n}$ in X is a Cauchy sequence if there exists a total function $g : N \to N$ such that, for all $l, m, n \in N$ , $\begin{array}{r} if l, m ⩾ g (n) then d (x_{l}, x_{m}) ⩽ 2^{- n} . \end{array}$ A function g with this property is called a Cauchy modulus of ${(x_{n})}_{n}$ .

The elementary proof of the following lemma is left to the reader.
Lemma 3.3.
Let $(X, d)$ be a metric space, Let ${(x_{n})}_{n}$ be a convergent sequence in X.
If a function $g : N \to N$ is a modulus of convergence of ${(x_{n})}_{n}$ then the function $h : N \to N$ defined by $h (n) : = g (n + 1)$ , for all $n \in N$ , is a Cauchy modulus of ${(x_{n})}_{n}$ .

The sequence ${(x_{n})}_{n}$ in X is a Cauchy sequence.

A Cauchy modulus of ${(x_{n})}_{n}$ is a modulus of convergence of ${(x_{n})}_{n}$ as well.

A metric space $(X, d)$ is called complete if every Cauchy sequence is convergent. Thus, a sequence ${(x_{n})}_{n}$ in a complete metric space $(X, d)$ is convergent if, and only if, it is a Cauchy sequence.

Based on the notions above we can now define computable versions of convergence and of the Cauchy condition.
Definition 3.4.
Let $(X, d)$ be a metric space.
A sequence ${(x_{n})}_{n}$ in X converges computably or is computably convergent if it converges and there exists a computable modulus $g : N \to N$ of convergence of ${(x_{n})}_{n}$ .

A sequence ${(x_{n})}_{n}$ in X is computably Cauchy if there exists a computable Cauchy modulus of ${(x_{n})}_{n}$ .

Corollary 3.5.
Let $(X, d)$ be a metric space.
If a sequence ${(x_{n})}_{n}$ in X converges computably then it is computably Cauchy.

If a sequence ${(x_{n})}_{n}$ in X is computably Cauchy and converges then it converges computably.

A sequence in a complete metric space converges computably if, and only if, it is computably Cauchy.

Proof.
The assertions follow from Lemma 3.3. □

We end this section with a well-known observation on Cauchy sequences in a normed vector space. A normed vector space X with norm $| \cdot |$ becomes a metric space with the metric $d : X^{2} \to R$ defined by $d (x, y) : = | x - y |$ , for $x, y \in X$ . A sequence ${(x_{n})}_{n}$ in a normed vector space X with norm $| \cdot |$ is called bounded if there exists a number $B \in N$ such that $| x_{n} | ⩽ B$ , for all $n \in N$ . If a number B with this property does not exist then ${(x_{n})}_{n}$ is called unbounded.
Lemma 3.6.
Any Cauchy sequence in a normed vector space is bounded.

We omit the very easy and well known proof. For example, for any $n ⩾ 1$ the set $R^{n}$ with the usual Euclidean norm $| (x_{1}, \dots, x_{n}) | : = \sqrt{\sum_{i = 1}^{n} x_{i}^{2}}$ is a normed vector space. In the case $n = 1$ the norm $| \cdot |$ is simply the usual absolute value function.
4. Computable sequences

First we have a look at computable sequences of rational numbers and computable sequences of real numbers. Then we shortly discuss computable sequences in computable metric spaces.

Definition 4.1.
A sequence ${(q_{n})}_{n}$ of rational numbers is called computable if there exists a total computable function $f : N \to N$ such that, for all $n \in N$ , $q_{n} = ν_{Q} (f (n))$ .
Definition 4.2.

A real number x is called computable if there exists a computable sequence ${(q_{n})}_{n}$ of rational numbers satisfying, for all $n \in N$ , $| x - q_{n} | ⩽ 2^{- n}$ . Let $EC$ be the set of all computable numbers.

A real number x is called left-computable if there exists an increasing computable sequence ${(q_{n})}_{n}$ of rational numbers converging to x. Let $LC$ be the set of all left-computable numbers.

Lemma 4.3.
For a real number x the following conditions are equivalent:
x is computable.

There exists a computable sequence ${(q_{n})}_{n}$ of rational numbers converging to x computably.

x and $- x$ are left-computable.

This is well-known and easy to see. We omit the proof.
Definition 4.4.
A sequence ${(x_{n})}_{n}$ of real numbers is called computable if there exists a computable sequence ${(q_{n})}_{n}$ of rational numbers such that, for all $k, n \in N$ , $| x_{n} - q_{⟨ n, k ⟩} | ⩽ 2^{- k}$ .

A sequence of rational numbers can be computable as a sequence of rational numbers, so, computable according to Definition 4.1, or it can be computable as a sequence of real numbers, hence, computable according to Definition 4.4. The following proposition clarifies the relation between these two types of sequences of rational numbers.
Proposition 4.5.

If a sequence ${(x_{n})}_{n}$ of rational numbers is a computable sequence of rational numbers as in Definition 4.1 then it is also a computable sequence of real numbers as in Definition 4.4 .

([ 25 , Exercise 4.2.7]) There exists a sequence ${(x_{n})}_{n}$ of rational numbers that is a computable sequence of real numbers according to Definition 4.4 but not a computable sequence of rational numbers according to Definition 4.1 .

We omit the simple proofs of the assertions. Later we shall need the following two observations about nondecreasing computable sequences of real numbers. The first one is certainly well known.
Lemma 4.6.
Let ${(a_{n})}_{n}$ be a nondecreasing computable sequence of real numbers converging to a computable real number. Then the sequence ${(a_{n})}_{n}$ converges computably.
Proof.
Let $α : = {lim}_{n \to \infty} a_{n}$ . Let ${(c_{n})}_{n}$ be a computable sequence of rational numbers such that, for all $n, k \in N$ , $| a_{n} - c_{⟨ n, k ⟩} | ⩽ 2^{- k}$ . Then the sequence ${(d_{n})}_{n}$ of rational numbers defined by $d_{n} : = c_{⟨ n, n + 2 ⟩}$ , for $n \in N$ , is computable as well and converges to α. Let ${(b_{n})}_{n}$ be a computable sequence of rational numbers with $| b_{n} - α | ⩽ 2^{- (n + 2)}$ , for all $n \in N$ . As both ${(d_{n})}_{n}$ and ${(b_{n})}_{n}$ converge to the same number and are computable, there exists a total, increasing, computable function $g : N \to N$ such $| d_{g (n)} - b_{g (n)} | ⩽ 2^{- (n + 2)}$ , for all $n \in N$ . We claim that g is a modulus of convergence of the sequence ${(a_{n})}_{n}$ . Indeed, for all $m, n \in N$ with $m ⩾ g (n)$ we obtain $\begin{aligned} | a_{m} - α | & ⩽ | a_{g (n)} - α | \\ ⩽ | a_{g (n)} - d_{g (n)} | + | d_{g (n)} - b_{g (n)} | + | b_{g (n)} - α | \\ ⩽ 2^{- (g (n) + 2)} + 2^{- (n + 2)} + 2^{- (g (n) + 2)} \\ ⩽ 2^{- (n + 2)} + 2^{- (n + 2)} + 2^{- (n + 2)} \\ < 2^{- n} . \end{aligned}$ □
Definition 4.7.
A function $s : N \to N$ tends to ∞ if for every $M \in N$ there exists some $N \in N$ such that for all $n \in N$ , $\begin{array}{r} if n ⩾ N then s (n) ⩾ M . \end{array}$

For example any unbounded nondecreasing function $s : N \to N$ tends to ∞.
Lemma 4.8.
Let ${(x_{n})}_{n}$ be a nondecreasing sequence of real numbers. Let $s : N \to N$ be a computable function that tends to ∞. If the sequence ${(x_{s (n + 1)} - x_{s (n)})}_{n}$ converges computably to 0 then the sequence ${(x_{n + 1} - x_{n})}_{n}$ converges computably to 0 as well.
Proof.
Let us assume that the sequence ${(x_{s (n + 1)} - x_{s (n)})}_{n}$ converges computably to 0. Let $g : N \to N$ be a computable modulus of convergence of the sequence ${(x_{s (n + 1)} - x_{s (n)})}_{n}$ . As this sequence converges to 0 this means that, for all $m, n \in N$ , $\begin{array}{r} m ⩾ g (n) \Rightarrow | x_{s (m + 1)} - x_{s (m)} | ⩽ 2^{- n} . \end{array}$ The function $s \circ g$ is total and computable. We claim that (A) the sequence ${(x_{n + 1} - x_{n})}_{n}$ converges to 0 and that (B) the function $s \circ g$ is a computable modulus of convergence of this sequence. Indeed, let us consider two numbers $k, n \in N$ satisfying $k ⩾ s (g (n))$ . Then the number $\begin{array}{r} m : = max {i \in N : s (i) ⩽ k} \end{array}$ exists because s tends to ∞. The number m satisfies $m ⩾ g (n)$ and $s (m) ⩽ k < s (m + 1)$ . We obtain $\begin{array}{r} x_{s (m)} ⩽ x_{k} ⩽ x_{k + 1} ⩽ x_{s (m + 1)}, \end{array}$ hence, $\begin{array}{r} | x_{k + 1} - x_{k} | ⩽ | x_{s (m + 1)} - x_{s (m)} | ⩽ 2^{- n} . \end{array}$ This shows the claims (A) and (B). □

Let us conclude this section with a short discussion of computable sequences and elements in a computable metric space.
Definition 4.9.
Let $(X, d)$ be a metric space, and let $α, β : N \to X$ be sequences in X.
We write $α ⩽ β$ and say that α is β-computable if there exists a total computable function $g : N^{2} \to N$ such that, for all $n, k \in N$ , $\begin{array}{r} d (α (n), β (g (n, k))) ⩽ 2^{- k} . \end{array}$

We write $α \equiv β$ and say that α and β are equivalent if $α ⩽ β$ and $β ⩽ α$ .

An element $x \in X$ is called α-computable if there exists a total computable function $f : N \to N$ such that, for all $k \in N$ , $d (x, α (f (k))) ⩽ 2^{- k}$ .

Note that the relation ⩽ is reflexive and transitive. Furthermore, $α ⩽ β$ implies that the set ${α (n) : n \in N}$ is a subset of the closure of the set ${β (n) : n \in N}$ . The relation ≡ is an equivalence relation. Obviously, an element $x \in X$ is α-computable iff the constant sequence $β : N \to X$ defined by $β (n) : = x$ , for all $n \in N$ , is α-computable.
Definition 4.10.
A computable metric space is a triple $(X, d, α)$ such that the pair $(X, d)$ is a metric space and $α : N \to X$ is a sequence such that the set ${α (n) : n \in N}$ is a dense subset of X and the function $n \mapsto d (α (π_{1} (n)), α (π_{2} (n)))$ is a computable sequence of real numbers.
Definition 4.11.
Let $(X, d, α)$ be a computable metric space. An element $x \in X$ (a sequence $γ : N \to X$ ) is called computable if it is α-computable.

The transitivity of ⩽ on sequences in X implies that in this definition α can be replaced by any equivalent sequence β in X without changing the set of computable elements of X or the set of computable sequences in X.
Lemma 4.12.
Let $(X, d, α)$ be a computable metric space. An element $x \in X$ is computable iff there exists a computable function $f : N \to N$ such that the sequence ${(α (f (n)))}_{n}$ converges computably to x.

We omit the simple proof.
5. Nearly computably Cauchy sequences in a metric space

The following definition is central for this paper.

Definition 5.1.
Let $(X, d)$ be a metric space. We say that a sequence ${(x_{n})}_{n}$ is nearly computably Cauchy if for every total, computable, and increasing function $r : N \to N$ the sequence of real numbers ${(d (x_{r (n + 1)}, x_{r (n)}))}_{n}$ converges computably to 0.

The motivation behind this definition is that for any convergent sequence, in fact, even for any Cauchy sequence, ${(x_{n})}_{n}$ the sequence ${(d (x_{n + 1}, x_{n}))}_{n}$ of distances of consecutive elements converges to 0. Of course, this applies to any subsequence as well. In the definition above we consider an effective version of the condition that the sequence of distances of consecutive elements of any effective subsequence converges to 0.
Lemma 5.2.
Let $(X, d)$ be a metric space. If a sequence ${(x_{n})}_{n}$ in X is computably Cauchy then it is nearly computably Cauchy.
Proof.
Let ${(x_{n})}_{n}$ be a sequence in X that is computably Cauchy. Let $r : N \to N$ be a total, increasing, computable function. We show that the sequence ${(d (x_{r (n + 1)}, x_{r (n)}))}_{n}$ converges computably to 0. Let $g : N \to N$ be a computable Cauchy modulus of ${(x_{n})}_{n}$ , that is, g is a total, computable function satisfying, for all $l, m, n \in N$ , $\begin{array}{r} l, m ⩾ g (n) \Rightarrow d (x_{l}, x_{m}) ⩽ 2^{- n} . \end{array}$ We claim that g shows that the sequence ${(d (x_{r (n + 1)}, x_{r (n)}))}_{n}$ converges computably to 0. Indeed, if $m ⩾ g (n)$ then $r (m + 1) ⩾ m + 1 ⩾ g (n)$ and $r (m) ⩾ m ⩾ g (n)$ , hence, $d (x_{r (m + 1)}, x_{r (m)}) ⩽ 2^{- n}$ . □

In the following lemma we show that one may modify the definition in several ways without changing the set of defined objects. In the lemma we consider for a total function $r : N \to N$ the following three conditions:
r is increasing,

r is nondecreasing and unbounded,

r is nondecreasing.
Note that (I) implies (II) and (II) implies (III).
Lemma 5.3.
Let $(X, d)$ be a metric space, and let ${(x_{n})}_{n}$ be a sequence in X. Then the three conditions
for every computable function $r : N \to N$ satisfying condition $(Y)$ the sequence ${(d (x_{r (n + 1)}, x_{r (n)}))}_{n}$ converges computably to 0
for $Y \in {I, II, III}$ are equivalent.
Proof.
The two implications “ $(II . C) \Rightarrow (I . C)$ ” and “ $(III . C) \Rightarrow (II . C)$ ” are trivial. It is therefore sufficient to prove the two implications “ $(I . C) \Rightarrow (II . C)$ ” and “ $(II . C) \Rightarrow (III . C)$ ”.

We prove “ $(I . C) \Rightarrow (II . C)$ ”. Let us assume that $(I . C)$ is true. Let $r : N \to N$ be a total, nondecreasing, unbounded computable function. We define a total function $f : N \to N$ recursively by $f (0) : = 0$ and $\begin{array}{r} f (n + 1) : = min {m \in N : r (m) > r (f (n))} \end{array}$ for all $n \in N$ . It is clear that f is well-defined, increasing and computable. Note that $\begin{array}{r} r (f (n)) = r (f (i)) = r (f (n + 1) - 1), \end{array}$ for all $n \in N$ and all $i \in {f (n), \dots, f (n + 1) - 1}$ . The function $s : = r \circ f : N \to N$ is a total, computable, and increasing function. By $(I . C)$ the sequence ${(d (x_{s (n + 1)}, x_{s (n)}))}_{n}$ converges computably to 0. Let $g : N \to N$ be a computable modulus of convergence of the sequence ${(d (x_{s (n + 1)}, x_{s (n)}))}_{n}$ , that is, $g : N \to N$ is a total, computable function such that, for all $m, n \in N$ , $\begin{array}{r} m ⩾ g (n) \Rightarrow d (x_{s (m + 1)}, x_{s (m)}) ⩽ 2^{- n} . \end{array}$ The function $h : = f \circ g$ is total and computable as well. We claim that it is a computable modulus of convergence of the sequence ${(d (x_{r (n + 1)}, x_{r (n)}))}_{n}$ and that this sequence converges to 0, that is, for all $m, n \in N$ $\begin{array}{r} m ⩾ h (n) \Rightarrow d (x_{r (m + 1)}, x_{r (m)}) ⩽ 2^{- n} . \end{array}$ Let $m ⩾ h (n)$ . As r is nondecreasing, either $r (m + 1) = r (m)$ or $r (m + 1) > r (m)$ . In the first case we clearly have $d (x_{r (m + 1)}, x_{r (m)}) = 0$ . In the second case there is some $k \in N$ with $m + 1 = f (k)$ . In fact, $k > 0$ as $f (0) = 0$ . We observe $r (m + 1) = r (f (k)) = s (k)$ and $r (m) = r (f (k) - 1) = r (f (k - 1)) = s (k - 1)$ . As f is increasing, from $\begin{array}{r} f (k) = m + 1 > m ⩾ h (n) = f (g (n)) \end{array}$ we conclude $k > g (n)$ , hence, $k - 1 ⩾ g (n)$ . Finally, we obtain $\begin{array}{r} d (x_{r (m + 1)}, x_{r (m)}) = d (x_{s (k)}, x_{s (k - 1)}) ⩽ 2^{- n} . \end{array}$ We have shown that $(I . C)$ implies $(II . C)$ .

Now, we prove “ $(II . C) \Rightarrow (III . C)$ ”. Let us assume that $(II . C)$ is true. Let $r : N \to N$ be a total, nondecreasing, computable function. If r is unbounded then the assertion, that ${(d (x_{r (n + 1)}, x_{r (n)}))}_{n}$ converges computably to 0, is true by assumption. Let us assume that r is bounded. Then r is ultimately constant, that is, there exists a number $N \in N$ such that $r (n) = r (N)$ , for all $n ⩾ N$ . Then $x_{r (n + 1)} = x_{r (n)}$ , for all $n ⩾ N$ . Hence, the constant function $g : N \to N$ defined by $g (n) : = N$ , for all $n \in N$ , shows that the sequence ${(d (x_{r (n + 1)}, x_{r (n)}))}_{n}$ converges computably to 0. □

In the following lemma we consider the special case of a normed vector space and show that in this case the condition in Definition 5.1 can be modified in further ways without changing the set of defined objects. In the following lemma for a sequence ${(y_{n})}_{n}$ in a normed vector space we consider the following three conditions:
${(y_{n})}_{n}$ is computably Cauchy,

${(y_{n})}_{n}$ converges computably,

${(y_{n})}_{n}$ converges computably to the zero in the normed vector space.
Note that (C) implies (B) and by Corollary 3.5.1 (B) implies (A). Note also that a sequence ${(y_{n})}_{n}$ converges computably to the zero in the normed vector space iff the sequence ${(| y_{n} |)}_{n}$ converges computably to the real number 0 (where for y in the normed vector space by $| y |$ we denote its norm).
Lemma 5.4.
Let X be a normed vector space with norm $| \cdot |$ , and let ${(x_{n})}_{n}$ be a sequence in X. Then the nine conditions
for every computable function $r : N \to N$ satisfying condition $(Y)$ the sequence ${(x_{r (n + 1)} - x_{r (n)})}_{n}$ satisfies condition $(Z)$ ,
for $Y \in {I, II, III}$ and $Z \in {A, B, C}$ are equivalent.
Proof.
The three implications “ $(II . Z) \Rightarrow (I . Z)$ ” and the three implications “ $(III . Z) \Rightarrow (II . Z)$ ”, for $Z \in {A, B, C}$ are trivial. The three implications “ $(Y . B) \Rightarrow (Y . A)$ ” for $Y \in {I, II, III}$ follow from Corollary 3.5.1, and the three implications “ $(Y . C) \Rightarrow (Y . B)$ ” for $Y \in {I, II, III}$ are trivial. The equivalence of the three conditions $(I . C)$ , $(II . C)$ , and $(III . C)$ follows from Lemma 5.3 because a sequence ${(x_{r (n + 1)} - x_{r (n)})}_{n}$ converges computably to the zero in X iff the sequence ${(| x_{r (n + 1)} - x_{r (n)} |)}_{n}$ converges computably to the real number 0.

It is therefore sufficient to prove the implication “ $(I . A) \Rightarrow (I . C)$ ”. Let us assume that $(I . A)$ is true. Let $r : N \to N$ be a total, increasing, computable function. We define $y_{n} : = x_{r (n + 1)} - x_{r (n)}$ , for all $n \in N$ . By $(I . A)$ the sequence ${(y_{n})}_{n}$ is computably Cauchy. We wish to show that it converges computably to $0 \in X$ . According to Corollary 3.5.2 it is sufficient to show that it converges to $0 \in X$ . For the sake of a contradiction, let us assume that it does not converge to 0. Then there exists a real number $ε > 0$ such that $| y_{n} | > ε$ for infinitely many n. Remember that the sequence ${(y_{n})}_{n}$ is computably Cauchy, hence, Cauchy. We choose a number $N \in N$ such that $| y_{N} | > ε$ and, for all $m, n ⩾ N$ , $| y_{m} - y_{n} | ⩽ ε / 2$ . We define $z : = y_{N}$ . Then $| z | > ε$ and, for all $n ⩾ N$ , $| y_{n} - z | ⩽ ε / 2$ . We define a total, increasing, and computable function $t : N \to N$ by $\begin{array}{r} t (n) : = r (\frac{n (n + 1)}{2}), \end{array}$ for all $n \in N$ . On the one hand, by (I.A) the sequence ${(x_{t (n + 1)} - x_{t (n)})}_{n}$ is computably Cauchy, hence, Cauchy. On the other hand, for all sufficiently large i, we have $\begin{array}{r} | x_{r (i + 1)} - x_{r (i)} - z | = | y_{i} - z | ⩽ ε / 2 . \end{array}$ We obtain for all sufficiently large n $\begin{aligned} | x_{t (n + 1)} - x_{t (n)} - (n + 1) \cdot z | & = | x_{r (\frac{(n + 1) (n + 2)}{2})} - x_{r (\frac{n (n + 1)}{2})} - (n + 1) \cdot z | \\ = | \sum_{i = 0}^{n} (x_{r (i + 1 + \frac{n (n + 1)}{2})} - x_{r (i + \frac{n (n + 1)}{2})}) - (n + 1) \cdot z | \\ ⩽ \sum_{i = 0}^{n} | x_{r (i + 1 + \frac{n (n + 1)}{2})} - x_{r (i + \frac{n (n + 1)}{2})} - z | \\ ⩽ (n + 1) \cdot ε / 2 \\ < \frac{n + 1}{2} \cdot | z | . \end{aligned}$ Hence, for all sufficiently large n we obtain $\begin{array}{r} | x_{t (n + 1)} - x_{t (n)} | ⩾ \frac{n + 1}{2} \cdot | z | . \end{array}$ Thus, the sequence ${(x_{t (n + 1)} - x_{t (n)})}_{n}$ is unbounded. This is in contradiction to the fact that the sequence ${(x_{t (n + 1)} - x_{t (n)})}_{n}$ is a Cauchy sequence; compare Lemma 3.6. We have shown that the sequence ${(x_{r (n + 1)} - x_{r (n)})}_{n}$ converges to 0. Thus, we have shown that $(I . A)$ implies $(I . C)$ . □

In Lemma 5.2 we have seen that any computably Cauchy sequence is nearly computably Cauchy. At the conference CCA 2018, after the presentation of [12], M. Schröder asked the following question: is every nearly computably Cauchy sequence necessarily a Cauchy sequence? Perhaps at least if it is an increasing sequence of real or rational numbers? It turns out that for the answer it is important whether ${(x_{n})}_{n}$ is computable or not.
Proposition 5.5.
Let $(X, d, α)$ be a computable metric space. Let ${(x_{n})}_{n}$ be a computable sequence in X. If it is nearly computably Cauchy then it is Cauchy.
Proof.
Let ${(x_{n})}_{n}$ be a computable sequence in X that is nearly computably Cauchy. We wish to show that it is a Cauchy sequence. For the sake of a contradiction, let us assume that this is not the case. Then there exists an integer $n_{0}$ such that for any $N \in N$ there are integers $k, l$ with $N ⩽ k < l$ and $d (x_{k}, x_{l}) > 5 \cdot 2^{- n_{0}}$ . There exists a computable function $f : N \to N$ such that $d (α_{f (n)}, x_{n}) < 2^{- n_{0}}$ , for all $n \in N$ . We construct a computable function $t : N \to N$ as follows. First, search for a pair of numbers $k, l$ with $k < l$ and $d (α_{f (k)}, α_{f (l)}) > 3 \cdot 2^{- n_{0}}$ and set $t (0) : = k$ and $t (1) : = l$ . And, for $n > 0$ , search for a pair of numbers $k, l$ with $t (2 n - 1) < k < l$ and $d (α_{f (k)}, α_{f (l)}) > 3 \cdot 2^{- n_{0}}$ and set $t (2 n) : = k$ and $t (2 n + 1) : = l$ . It is clear that in this way an increasing and computable function $t : N \to N$ is obtained that satisfies, for all $n \in N$ , $d (α_{f (t (2 n))}, α_{f (t (2 n + 1))}) > 3 \cdot 2^{- n_{0}}$ , hence, $d (x_{t (2 n)}, x_{t (2 n + 1)}) > 2^{- n_{0}}$ . But, as ${(x_{n})}_{n}$ is nearly computably Cauchy, the sequence ${(d (x_{t (n + 1)}, x_{t (n)}))}_{n}$ should converge to 0. Contradiction. □

In particular, if a computable sequence of real numbers is nearly computably Cauchy then it converges. But in the general case the answer to the question formulated before the previous proposition is no.
Theorem 5.6.
There exists an increasing and unbounded sequence of rational numbers that is nearly computably Cauchy.

Note that by Lemma 3.6 an unbounded sequence cannot be a Cauchy sequence, hence, it cannot be convergent. For the proof of this theorem we first formulate a lemma. For this lemma we need some further well-known computability-theoretic notions. The notion of a computable function $H : \subseteq N^{N} \to N^{N}$ is used in the usual sense; for a precise definition see, e.g., [20, Section 9.2.3]. Let $ρ_{Q^{N}} : N^{N} \to Q^{N}$ be defined by $ρ_{Q^{N}} (p) : = (ν_{Q} (p_{0}), ν_{Q} (p_{1}), ν_{Q} (p_{2}), \dots)$ , for $p = {(p_{i})}_{i} \in N^{N}$ . A function $F : \subseteq Q^{N} \times N \times N^{N} \to Q^{N}$ is called computable if there exists a computable function $H : \subseteq N^{N} \to N^{N}$ with $\begin{aligned} dom (H) = {p \in N^{N} : (ρ_{Q^{N}} (p_{1} p_{3} p_{5} \dots), p_{0}, (p_{2} p_{4} p_{6} \dots)) \in dom (F)} \\ ρ_{Q^{N}} (H (p)) = F (ρ_{Q^{N}} (p_{1} p_{3} p_{5} \dots), p_{0}, (p_{2} p_{4} p_{6} \dots)), \end{aligned}$ for $p \in dom (H)$ . Similarly, the notion of a computable function $G : \subseteq Q^{N} \times N \times N^{N} \to N^{N}$ is defined.
Lemma 5.7.
There exist computable functions $F : \subseteq Q^{N} \times N \times N^{N} \to Q^{N}$ and $G : \subseteq Q^{N} \times N \times N^{N} \to N^{N}$ such that for any triple $(a, N, s)$ consisting of
an increasing, unbounded sequence $a : N \to Q$ of rational numbers,

a natural number N, and

an increasing function $s : N \to N$
both $b : = F (a, N, s)$ and $g : = G (a, N, s)$ are defined and satisfy the following conditions:
$b : N \to Q$ is an increasing, unbounded sequence of rational numbers,

for all $k \in N$ , if $a_{k} ⩽ N$ then $b_{k} = a_{k}$ ,

for all $k, l \in N$ , if $k ⩽ l$ then $b_{l} - b_{k} ⩽ a_{l} - a_{k}$ ,

the sequence ${(b_{s (k + 1)} - b_{s (k)})}_{k}$ converges to 0 and the function $g : N \to N$ is a modulus of convergence of this sequence.

Conditions (II) and (III) mean that the sequence b is obtained from the sequence a by “compressing” the part of a with entries larger than N and by leaving the other part unchanged.
Proof.
Let an increasing, unbounded sequence $a : N \to Q$ of rational numbers, a natural number N, and an increasing function $s : N \to N$ be given. We construct the desired sequence $b : N \to Q$ and the desired function $g : N \to N$ step by step. Note that we will ensure that $b_{s (g (n))} ⩾ N + n$ , for all n. This implies that the new sequence ${(b_{n})}_{n}$ will be unbounded. Furthermore, the constructed function g will be increasing. First, we define $\begin{array}{r} g (0) : = min {k \in N : a_{s (k)} ⩾ N} \end{array}$ and $b_{j} : = a_{j}$ , for all $j ⩽ s (g (0))$ . Note that this ensures that $b_{s (g (0))} ⩾ N$ and that Condition (II) is satisfied. Let us assume that for some $n \in N$ we have already defined $g (0), \dots, g (n)$ and $b_{0}, \dots, b_{s (g (n))}$ such that
$g (0) < \dots < g (n)$ ,

$b_{0} < \dots < b_{s (g (n))}$ ,

for all $m ⩽ n$ , $b_{s (g (m))} ⩾ N + m$ ,

for all $k, l ⩽ s (g (n))$ , if $k ⩽ l$ then $b_{l} - b_{k} ⩽ a_{l} - a_{k}$ ,

for all $m < n$ and all i with $g (m) ⩽ i < g (n)$ , $b_{s (i + 1)} - b_{s (i)} ⩽ 2^{- m}$ .
We define $\begin{array}{r} g (n + 1) : = min {k \in N : k > g (n) and \sum_{i = g (n)}^{k - 1} min {2^{- n}, a_{s (i + 1)} - a_{s (i)}} ⩾ 1} . \end{array}$ This value $g (n + 1)$ is well-defined because the sequence a is unbounded. For $i = g (n), \dots, g (n + 1) - 1$ and $j = s (i) + 1, \dots, s (i + 1)$ we define $\begin{array}{r} b_{j} : = \{\begin{array}{ll} b_{j - 1} + (a_{j} - a_{j - 1}) & if a_{s (i + 1)} - a_{s (i)} ⩽ 2^{- n}, \\ b_{j - 1} + (a_{j} - a_{j - 1}) \cdot \frac{2^{- n}}{a_{s (i + 1)} - a_{s (i)}} & otherwise . \end{array} \end{array}$ We have defined $b_{s (g (n)) + 1}, \dots, b_{s (g (n + 1))}$ . The construction is illustrated in Fig. 1.
Figure 1.
An illustration of the construction in the proof of Lemma 5.7.

It is clear that $b_{0} < \dots < b_{s (g (n + 1))}$ and that, for all $k, l ⩽ s (g (n + 1))$ , if $k ⩽ l$ then $b_{l} - b_{k} ⩽ a_{l} - a_{k}$ . Furthermore, for $i \in {g (n), \dots, g (n + 1) - 1}$ , $\begin{array}{r} b_{s (i + 1)} - b_{s (i)} = \{\begin{array}{ll} a_{s (i + 1)} - a_{s (i)} & if a_{s (i + 1)} - a_{s (i)} ⩽ 2^{- n}, \\ 2^{- n} & otherwise . \end{array} \end{array}$ Finally, $\begin{aligned} b_{s (g (n + 1))} & = b_{s (g (n))} + b_{s (g (n + 1))} - b_{s (g (n))} \\ = b_{s (g (n))} + \sum_{i = g (n)}^{g (n + 1) - 1} (b_{s (i + 1)} - b_{s (i)}) \\ ⩾ N + n + \sum_{i = g (n)}^{g (n + 1) - 1} min {2^{- n}, a_{s (i + 1)} - a_{s (i)}} \\ ⩾ N + n + 1 . \end{aligned}$ □
Proof of Theorem 5.6.
We define a sequence ${(q_{n})}_{n}$ of rational numbers as follows. Let F and G be computable functions as in Lemma 5.7. Let ${(s^{(i)})}_{i}$ be a sequence consisting of all total, increasing, computable functions $s^{(i)} : N \to N$ (of course, this sequence is not computable). We define a sequence ${(a^{(i)})}_{i}$ of sequences $a^{(i)} : N \to Q$ of rational numbers as follows. Let $a^{(0)}$ be defined by $a_{n}^{(0)} : = n$ , for all n. For all $i \in N$ we define $\begin{array}{r} a^{(i + 1)} : = F (a^{(i)}, i + 1, s^{(i)}) . \end{array}$ The sequence $a^{(0)}$ is an increasing, unbounded, computable sequence of rational numbers. As F is computable, it follows by induction that, for all $i \in N$ , the sequence $a^{(i)}$ is an increasing, unbounded, computable sequence of rational numbers. Finally, we define $q_{n} : = {lim}_{i \to \infty} a_{n}^{(i)}$ , for all n. We claim that the sequence $q : N \to Q$ is well-defined and has the desired properties. First, we show that q is well-defined and, for all $i, n \in N$ , $\begin{aligned} (1) & if n ⩽ i then q_{n} = a_{n}^{(i)} . \end{aligned}$ As $a_{0}^{(0)} = 0 < j$ , for all $j > 0$ , by Condition (II) in Lemma 5.7 we obtain $a_{0}^{(j)} = 0$ for all $j \in N$ , by induction. Let us consider some n. Due to Condition (III) in Lemma 5.7, for all $j \in N$ , $a_{n}^{(j)} = a_{n}^{(j)} - a_{0}^{(j)} ⩽ a_{n}^{(0)} - a_{0}^{(0)} = n < n + 1$ , for all $j \in N$ . But then, by Condition (II) in Lemma 5.7, $a_{n}^{(i + 1)} = a_{n}^{(i)}$ , for all $i ⩾ n$ . This implies that $q = {lim}_{i \to \infty} a_{n}^{(i)}$ is well-defined, for all $n \in N$ , and that (1) is true. The sequence q is increasing because, for all $n \in N$ , $\begin{array}{r} q_{n + 1} - q_{n} = a_{n + 1}^{(n + 1)} - a_{n}^{(n + 1)} > 0 . \end{array}$ Next, we show that q is unbounded. For an arbitrary i, let n be the largest number with $a_{n}^{(i)} ⩽ i + 1$ . Then, by Condition (II) in Lemma 5.7, $a_{n}^{(j)} = a_{n}^{(i)}$ , for all $j ⩾ i$ , hence, by (1), $q_{n} = a_{n}^{(max {n, i})} = a_{n}^{(i)}$ and, by Condition (III) in Lemma 5.7, $\begin{array}{r} a_{n + 1}^{(i)} - a_{n}^{(i)} ⩽ a_{n + 1}^{(0)} - a_{n}^{(0)} = n + 1 - n = 1, \end{array}$ hence, $q_{n} = a_{n}^{(i)} ⩾ a_{n + 1}^{(i)} - 1 > i + 1 - 1 = i$ . Finally, for all $i \in N$ the function $g^{(i)} : = G (a^{(i)}, i + 1, s^{(i)})$ is computable because G is computable. We claim that it is a modulus of convergence to 0 of the sequence ${(q_{s^{(i)} (k + 1)} - q_{s^{(i)} (k)})}_{k}$ . By Condition (IV) in Lemma 5.7 the function $g^{(i)}$ is a modulus of convergence to 0 of the sequence ${(a_{s^{(i)} (k + 1)}^{(i)} - a_{s^{(i)} (k)}^{(i)})}_{k}$ . We show that this carries over from $a^{(i)}$ to q. Indeed, let us consider some $m, n \in N$ with $m ⩾ g^{(i)} (n)$ . We choose some $k ⩾ max {s^{(i)} (m + 1), i}$ . By (1) and by Condition (III) in Lemma 5.7 we obtain $\begin{array}{r} q_{s^{(i)} (m + 1)} - q_{s^{(i)} (m)} = a_{s^{(i)} (m + 1)}^{(k)} - a_{s^{(i)} (m)}^{(k)} ⩽ a_{s^{(i)} (m + 1)}^{(i)} - a_{s^{(i)} (m)}^{(i)} ⩽ 2^{- n} . \end{array}$ We have shown that q is an increasing, unbounded sequence of rational numbers satisfying Condition (I.C) in Lemma 5.3, thus, it is nearly computably Cauchy. □

In the rest of this section we make several useful observations about nearly computably Cauchy sequences in a metric space or a normed space and about nondecreasing sequences of real numbers that are nearly computably Cauchy. The following lemma will be used several times for sequences ${(x_{n})}_{n}$ and ${(y_{n})}_{n}$ satisfying even $d (x_{n}, y_{n}) ⩽ 2^{- n}$ , for all $n \in N$ .
Lemma 5.8.
Let $(X, d)$ be a metric space. Let ${(x_{n})}_{n}$ and ${(y_{n})}_{n}$ be sequences in X such that the sequence of real numbers ${(d (x_{n}, y_{n}))}_{n}$ converges computably to 0. Then the sequence ${(x_{n})}_{n}$ is nearly computably Cauchy if, and only if, the sequence ${(y_{n})}_{n}$ is nearly computably Cauchy.
Proof.
Let us assume that ${(x_{n})}_{n}$ is nearly computably Cauchy. By symmetry it is sufficient to show that this implies that ${(y_{n})}_{n}$ is nearly computably Cauchy as well. Let $f : N \to N$ be a computable modulus of convergence of the sequence ${(d (x_{n}, y_{n}))}_{n}$ . Let $r : N \to N$ be an increasing, computable function. Then the sequence ${(d (x_{r (n + 1)}, x_{r (n)}))}_{n}$ converges computably to 0. Let $g : N \to N$ be a computable modulus of convergence of this sequence. The function $h : N \to N$ defined by $h (n) : = max {f (n + 2), g (n + 2)}$ , for $n \in N$ , is computable. We claim that the sequence ${(d (y_{r (n + 1)}, y_{r (n)}))}_{n}$ converges to 0 and that h is a modulus of convergence of this sequence. Indeed, for $m, n \in N$ with $m ⩾ h (n)$ we obtain $r (m + 1) ⩾ m + 1 > h (n) ⩾ f (n + 2)$ and $m ⩾ h (n) ⩾ g (n + 2)$ as well as $r (m) ⩾ m ⩾ h (n) ⩾ f (n + 2)$ , hence, $\begin{aligned} d (y_{r (m + 1)}, y_{r (m)}) & ⩽ d (y_{r (m + 1)}, x_{r (m + 1)}) + d (x_{r (m + 1)}, x_{r (m)}) + d (x_{r (m)}, y_{r (m)}) \\ ⩽ 2^{- (n + 2)} + 2^{- (n + 2)} + 2^{- (n + 2)} \\ < 2^{- n} . \end{aligned}$ □
Lemma 5.9.
Let $(X, d)$ be a metric space, and let ${(x_{n})}_{n}$ be a sequence in X. Let $s : N \to N$ be a total, nondecreasing, unbounded computable function, and define $y_{n} : = x_{s (n)}$ , for all n. If ${(x_{n})}_{n}$ is nearly computably Cauchy then so is ${(y_{n})}_{n}$ .
Proof.
Let ${(x_{n})}_{n}$ be nearly computably Cauchy. If $r : N \to N$ is a total, nondecreasing, unbounded computable function then $s \circ r : N \to N$ is a total, nondecreasing, unbounded, computable function as well. Hence, the sequence $\begin{array}{r} {(d (y_{r (n + 1)}, y_{r (n)}))}_{n} = {(d (x_{s (r (n + 1))}, x_{s (r (n))}))}_{n} = {(d (x_{(s \circ r) (n + 1)}, x_{(s \circ r) (n)}))}_{n} \end{array}$ converges computably. □
Proposition 5.10.
The set of all nearly computably Cauchy sequences in a normed vector space is a sub vector space of the vector space of all sequences in the normed space.
Proof.
Let X be a normed vector space. We need to prove the following two assertions:
If ${(x_{n})}_{n}$ and ${(y_{n})}_{n}$ are nearly computably Cauchy sequences then ${(x_{n} + y_{n})}_{n}$ is a nearly computably Cauchy sequence as well.

If ${(x_{n})}_{n}$ is a nearly computably Cauchy sequence and λ is a real number then ${(λ \cdot x_{n})}_{n}$ is a nearly computably Cauchy sequence.
We prove the first assertion and leave the similar proof of the second assertion to the reader.

Let ${(x_{n})}_{n}$ and ${(y_{n})}_{n}$ be nearly computably Cauchy sequences. Let $r : N \to N$ be a computable and increasing function. We claim that the sequence ${(x_{r (n + 1)} + y_{r (n + 1)} - x_{r (n)} - y_{r (n)})}_{n}$ converges computably to 0. It is clear that it converges to 0 because the sequences ${(x_{r (n + 1)} - x_{r (n)})}_{n}$ and ${(y_{r (n + 1)} - y_{r (n)})}_{n}$ converge to 0 computably. Let $g_{x} : N \to N$ be a computable modulus of convergence of the sequence ${(x_{r (n + 1)} - x_{r (n)})}_{n}$ , and let $g_{y} : N \to N$ be a computable modulus of convergence of the sequence ${(y_{r (n + 1)} - y_{r (n)})}_{n}$ . Then the function $h : N \to N$ defined by $h (n) : = max {g_{x} (n + 1), g_{y} (n + 1)}$ is computable. It is a modulus of convergence of the sequence ${(x_{r (n + 1)} + y_{r (n + 1)} - x_{r (n)} - y_{r (n)})}_{n}$ because, for all $m, n \in N$ with $m ⩾ h (n)$ , $\begin{aligned} | x_{r (m + 1)} + y_{r (m + 1)} - x_{r (m)} - y_{r (m)} | & ⩽ | x_{r (m + 1)} - x_{r (m)} | + | y_{r (m + 1)} - y_{r (m)} | \\ ⩽ 2^{- (n + 1)} + 2^{- (n + 1)} \\ = 2^{- n} . \end{aligned}$ We have shown that the sequence ${(x_{r (n + 1)} + y_{r (n + 1)} - x_{r (n)} - y_{r (n)})}_{n}$ converges computably to 0. □
Proposition 5.11.
Let ${(x_{n})}_{n}$ and ${(y_{n})}_{n}$ be convergent nondecreasing computable sequences of real numbers with the same limit. Then ${(x_{n})}_{n}$ is nearly computably Cauchy if, and only if, ${(y_{n})}_{n}$ is nearly computably Cauchy.
Proof.
By symmetry it is sufficient to show one implication. So, let us assume that ${(x_{n})}_{n}$ is nearly computably Cauchy. We wish to show that ${(y_{n})}_{n}$ is nearly computably Cauchy. First we show that we can switch from ${(x_{n})}_{n}$ and ${(y_{n})}_{n}$ to increasing computable sequences of rational numbers. There exists a computable sequence ${(q_{n})}_{n}$ of rational numbers such that, for all $k, n \in N$ , $| x_{n} - q_{⟨ n, k ⟩} | ⩽ 2^{- k}$ . We define a sequence ${(a_{n})}_{n}$ of rational numbers by $a_{n} : = q_{⟨ n, n + 3 ⟩} - 5 \cdot 2^{- (n + 3)}$ , for $n \in N$ . It is clear that ${(a_{n})}_{n}$ is a computable sequence of rational numbers with $0 < 2^{- (n + 1)} ⩽ x_{n} - a_{n} ⩽ 3 \cdot 2^{- (n + 2)}$ , for all n. It is increasing because $\begin{aligned} a_{n} & = q_{⟨ n, n + 3 ⟩} - 5 \cdot 2^{- (n + 3)} \\ ⩽ x_{n} + 2^{- (n + 3)} - 5 \cdot 2^{- (n + 3)} \\ = x_{n} - 2^{- (n + 1)} \\ ⩽ x_{n + 1} - 2^{- (n + 1)} \\ < x_{n + 1} - 2^{- (n + 4)} - 5 \cdot 2^{- (n + 4)} \\ ⩽ q_{⟨ n + 1, n + 4 ⟩} - 5 \cdot 2^{- (n + 4)} \\ = a_{n + 1}, \end{aligned}$ for all n. In the same way we can conclude that there exists a computable increasing sequence ${(b_{n})}_{n}$ of rational numbers with $0 < 2^{- (n + 1)} ⩽ y_{n} - b_{n} ⩽ 3 \cdot 2^{- (n + 2)}$ , for all n.

According to Lemma 5.8 the sequence ${(a_{n})}_{n}$ is nearly computably Cauchy. The main step is to show that this implies that ${(b_{n})}_{n}$ is nearly computably Cauchy as well. Once that is done, by Lemma 5.8 the sequence ${(y_{n})}_{n}$ is nearly computably Cauchy as well.

So, we consider two increasing computable sequences ${(a_{n})}_{n}$ and ${(b_{n})}_{n}$ of rational numbers converging to the same limit such that ${(a_{n})}_{n}$ is nearly computably Cauchy, and we wish to show that ${(b_{n})}_{n}$ is nearly computably Cauchy as well. Let $r : N \to N$ be an arbitrary increasing computable function. It is sufficient to show that the sequence ${(b_{r (n + 1)} - b_{r (n)})}_{n}$ converges computably to 0. The sequence ${(c_{n})}_{n}$ defined by $c_{n} : = b_{r (n)}$ , for $n \in N$ , is computable, increasing and converges to the same number as ${(a_{n})}_{n}$ . It is sufficient to show that the sequence ${(c_{n + 1} - c_{n})}_{n}$ converges computably to 0. In order to show this we define two increasing computable functions $s, t : N \to N$ so that the sequences ${(a_{s (n)})}_{n}$ and ${(c_{t (n)})}_{n}$ overtake each other in turns. Let $s, t : N \to N$ be defined by $s (0) : = 0$ , by $t (0) : = min {k \in N : c_{k} > a_{0}}$ , and, for $n \in N$ , by $\begin{aligned} s (n + 1) : = min {k \in N : k > s (n) and a_{k} > c_{t (n)}}, \\ t (n + 1) : = min {k \in N : k > t (n) and c_{k} > a_{s (n + 1)}} . \end{aligned}$ It is clear that both functions s and t are increasing and computable and satisfy $\begin{array}{r} a_{s (n)} < c_{t (n)} < a_{s (n + 1)}, \end{array}$ for all $n \in N$ . As ${(a_{n})}_{n}$ is nearly computably Cauchy the sequence ${(a_{s (n + 1)} - a_{s (n)})}_{n}$ converges computably to 0. Let $g : N \to N$ be a modulus of convergence of this sequence. We claim that the computable function $h : N \to N$ defined by $h (n) : = g (n + 1)$ , for $n \in N$ , is a modulus of convergence of the sequence ${(c_{t (n + 1)} - c_{t (n)})}_{n}$ and that this sequence converges to 0. Indeed, for $m, n \in N$ with $m ⩾ h (n)$ we obtain $\begin{aligned} | c_{t (m + 1)} - c_{t (m)} | & = c_{t (m + 1)} - c_{t (m)} \\ < a_{s (m + 2)} - a_{s (m)} \\ = a_{s (m + 2)} - a_{s (m + 1)} + a_{s (m + 1)} - a_{s (m)} \\ ⩽ 2^{- (n + 1)} + 2^{- (n + 1)} \\ = 2^{- n} . \end{aligned}$ We have shown that the sequence ${(c_{t (n + 1)} - c_{t (n)})}_{n}$ converges computably to 0. By Lemma 4.8 the sequence ${(c_{n + 1} - c_{n})}_{n}$ converges computably to 0 as well. □

The following corollary of Proposition 5.11 shows that the converse of Lemma 5.9 is true for nondecreasing computable sequences of real numbers.
Corollary 5.12.
Let ${(x_{n})}_{n}$ be a nondecreasing computable sequence of real numbers. Let $t : N \to N$ be a total, nondecreasing, unbounded computable function, and define $y_{n} : = x_{t (n)}$ . Then ${(x_{n})}_{n}$ is nearly computably Cauchy if, and only if, ${(y_{n})}_{n}$ is nearly computably Cauchy.
6. The field of nearly computable real numbers

In this section we consider the set of nearly computable real numbers and their closure properties. Actually, “nearly computable” elements can be defined in any computable metric space.

Definition 6.1.
Let $(X, d, α)$ be a computable metric space. We call an element $y \in X$ nearly computable if there exists a computable sequence ${(x_{n})}_{n}$ in X that is nearly computably Cauchy and converges to y.

In this definition, α can be replaced by any equivalent (see the end of Section 4) sequence β in X without changing the set of nearly computable elements of X.
Proposition 6.2.
Let $(X, d, α)$ be a computable metric space. Every computable element of X is nearly computable.
Proof.
This follows from Lemma 4.12, Corollary 3.5.1, and Lemma 5.2. □

By applying this definition to the computable metric space $(R, d, ν_{Q})$ where d is the usual Euclidean distance, we obtain the notion of a nearly computable real number. Let $NC$ be the set of all nearly computable real numbers.

We say that a subset $S \subseteq R$ is closed under a function $f : \subseteq R^{k} \to R$ if for any vector $x \in dom (f) \cap S^{k}$ the real number $f (x)$ is an element of S. For $k ⩾ 1$ and for $i, j \in N$ we write $\begin{array}{r} B_{⟨ i, j ⟩}^{(k)} : = {x \in R^{k} : d (x, ν_{Q}^{(k)} (i)) < 2^{- j}} . \end{array}$ The following notion of a computable function $f : \subseteq R^{k} \to R$ can be characterized in many different ways; compare [25].
Definition 6.3.
A function $f : \subseteq R^{k} \to R$ is called computable if there exists a computably enumerable subset $A \subseteq N$ such that
for all $m, n \in N$ with $⟨ m, n ⟩ \in A$ , $f (B_{m}^{(k)}) \subseteq B_{n}^{(1)}$ ,

for all $x \in dom (f)$ and all $ε > 0$ there exist $m, n \in N$ with $x \in B_{m}^{(k)}$ , $⟨ m, n ⟩ \in A$ , and $2^{- π_{2} (n)} < ε$ .

The following characterization of computable functions on a compact rectangle with rational vertices is a minor straightforward generalization of a result by Grzegorczyk [8]. We omit the proof.
Lemma 6.4.
Let $k ⩾ 1$ . Let $a_{1}, \dots, a_{k}, b_{1}, \dots, b_{k}$ be rational numbers with $a_{i} < b_{i}$ , for $i = 1, \dots, k$ . Then a function $f : [a_{1}, b_{1}] \times \dots \times [a_{k}, b_{k}] \to R$ is computable if, and only if,
for any computable sequence ${(q_{n})}_{n}$ of rational points in $(a_{1}, b_{1}) \times \dots \times (a_{k}, b_{k})$ (that is, for any sequence ${(q_{n})}_{n} \in Q^{k} \cap (a_{1}, b_{1}) \times \dots \times (a_{k}, b_{k})$ such that there is a computable function $f : N \to N$ with $q_{n} = ν_{Q}^{(k)} (f (n))$ , for all n) the sequence ${(f (q_{n}))}_{n}$ is a computable sequence of real numbers,

there exists a computable function $g : N \to N$ such that, for all $n \in N$ and all $x, y \in [a_{1}, b_{1}] \times \dots \times [a_{k}, b_{k}]$ , if $d (x, y) ⩽ 2^{- g (n)}$ then $| f (x) - f (y) | ⩽ 2^{- n}$ (such a function g will be called a computable modulus of continuity of f).

Proposition 6.5.
The set $NC$ is closed under computable functions $f : \subseteq R^{k} \to R$ with open domain $dom (f) \subseteq R^{k}$ , for arbitrary $k > 0$ .
Proof.
Let $x_{1}, \dots, x_{k}$ be nearly computable numbers, and let $f : \subseteq R^{k} \to R$ be a computable function with open domain and with $(x_{1}, \dots, x_{k}) \in dom (f)$ . In the following we simply write x for the vector $(x_{1}, \dots, x_{k})$ . We wish to show that $f (x)$ is a nearly computable number. Let $a_{1}, \dots, a_{k}, b_{1}, \dots, b_{k}$ be rational numbers with $a_{i} < x_{i} < b_{i}$ , for $i = 1, \dots, k$ , and with $[a_{1}, b_{1}] \times \dots \times [a_{k}, b_{k}] \subseteq dom (f)$ . The restriction of f to the cube $[a_{1}, b_{1}] \times \dots \times [a_{k}, b_{k}]$ is computable as well. Let $g : N \to N$ be a computable modulus of continuity of this restriction; compare Lemma 6.4. Let ${(q_{n})}_{n}$ be a computable sequence of rational points that converges to x and is nearly computably Cauchy. We can assume that $q_{n} \in (a_{1}, b_{1}) \times \dots \times (a_{k}, b_{k})$ , for all n. If this should not be the case then we can achieve it by cutting off a finite prefix of the sequence ${(q_{n})}_{n}$ so that the resulting sequence satisfies this condition. The resulting sequence will still be computable, will still converge to x and will still be nearly computably Cauchy. By Lemma 6.4 the sequence ${(f (q_{n}))}_{n}$ is a computable sequence of real numbers that converges to $f (x)$ . We claim that it is nearly computably Cauchy. Let $r : N \to N$ be an increasing, computable function. Let $h : N \to N$ be a computable modulus of convergence of the sequence $d (q_{r (n + 1)}, q_{r (n)})$ . We claim that the computable function $h \circ g$ is a modulus of convergence of the sequence $| f (q_{r (n + 1)}) - f (q_{r (n)}) |$ . Indeed, for $m, n \in N$ with $m ⩾ h (g (n))$ we obtain $d (q_{r (m + 1)}, q_{r (m)}) ⩽ 2^{- g (n)}$ , hence $| f (q_{r (m + 1)}) - f (q_{r (m)}) | ⩽ 2^{- n}$ . Thus, the sequence ${(f (q_{n}))}_{n}$ is a computable sequence of real numbers that converges to $f (x)$ and is nearly computably Cauchy. Finally, let ${(p_{n})}_{n}$ be a computable sequence of rational numbers with $| f (q_{n}) - p_{n} | ⩽ 2^{- n}$ , for all n. Then this sequence converges to $f (x)$ as well. By Lemma 5.8 it is nearly computably Cauchy as well. This shows that $f (x)$ is a nearly computable number. □

A subset $S \subseteq R$ is called a subfield of $R$ if $0, 1 \in S$ and if S is closed under the three total functions addition $+ : R^{2} \to R$ , additive inverse $- : R \to R$ , and multiplication $\cdot : R^{2} \to R$ and under the partial function multiplicative inverse $1 / \cdot : R ∖ {0} \to R$ . A subset $S \subseteq R$ is called a real closed field if it is a subfield of $R$ and additionally for every $k > 0$ and for every nonzero vector of real numbers $b_{0}, \dots, b_{k} \in S$ (“nonzero vector” means that at least one of the numbers $b_{0}, \dots, b_{k}$ must be different from zero) every real number x with $b_{k} \cdot x^{k} + \dots + b_{1} \cdot x + b_{0} = 0$ is an element of S, that is, any real zero of any nonzero polynomial with coefficients in S is again an element of S.

Let $(X, d_{X})$ and $(Y, d_{Y})$ be metric spaces. A function $f : X \to Y$ is called Lipschitz continuous if there exists a constant $c > 0$ such that, for all $x, x^{'} \in X$ , $\begin{array}{r} d_{Y} (f (x), f (x^{'})) ⩽ c \cdot d_{X} (x, x^{'}) . \end{array}$ Such a constant c is then called a Lipschitz constant of f. It is well known that all norms on $R^{k}$ are equivalent. Therefore, a function $f : \subseteq R^{k} \to R$ is Lipschitz continuous with respect to the distance function induced by one norm if, and only if, it is Lipschitz continuous with respect to the distance function induced by any other norm. The following lemma is a slight strengthening of an observation implicitly contained in the article [18] by Raichev.
Lemma 6.6.
If a subset $K \subseteq R$ contains a number $x_{0} \neq 0$ and is closed under Lipschitz continuous computable functions $f : \subseteq R^{k} \to R$ with open domain $dom (f) \subseteq R^{k}$ , k arbitrary, then K is a real closed subfield of $R$ .
Proof.
On $R^{k}$ let us consider the maximum norm $‖ \cdot ‖_{\infty} : R^{k} \to R^{k}$ defined by $\begin{array}{r} {‖ (x_{1}, \dots, x_{k}) ‖}_{\infty} : = max {| x_{1} |, \dots, | x_{k} |}, \end{array}$ for any vector $(x_{1}, \dots, x_{k}) \in R^{k}$ .

The additive inverse $- : R \to R$ is a total, Lipschitz continuous, computable function with Lipschitz constant 1. The addition $+ : R^{2} \to R$ is a total, Lipschitz continuous, computable function with Lipschitz constant 2. For any two numbers $x, x^{'} \in R$ let us choose an integer N with $N > | x |$ and $N > | x^{'} |$ . The restriction $\cdot |_{{(- N, N)}^{2}} : {(- N, N)}^{2} \to R$ of the multiplication function to the open cube ${(- N, N)}^{2}$ is a Lipschitz continuous and computable function. For any number $x \in R ∖ {0}$ let us choose an integer $N > 1$ with $N > | x |$ and $1 / N < | x |$ . The restriction $\cdot |_{(- N, - 1 / N) \cup (1 / N, N)} : (- N, - 1 / N) \cup (1 / N, N) \to R$ of the multiplicative inverse function to the union of the two open intervals $(- N, - 1 / N)$ and $(1 / N, N)$ is a Lipschitz continuous and computable function. This shows that K is closed under the three total functions addition $+ : R^{2} \to R$ , additive inverse $- : R \to R$ , and multiplication $\cdot : R^{2} \to R$ and under the partial function multiplicative inverse $1 / \cdot : R ∖ {0} \to R$ . Let $x_{0}$ be an element of K with $x_{0} \neq 0$ . Then $0 = x_{0} + (- x_{0}) \in K$ and $1 = x_{0} \cdot (1 / x_{0}) \in K$ . We have shown that K is a subfield of $R$ .

Let $z \in K$ be a real number such that there exists a nonzero vector of real numbers $b_{0}, \dots, b_{k} \in K$ with $b_{k} \cdot z^{k} + \dots + b_{1} \cdot z + b_{0} = 0$ . Then $k > 0$ . We wish to show that z is an element of K as well. Let $p (x)$ be the polynomial $p (x) : = \sum_{i = 0}^{k} b_{i} \cdot x^{i}$ . In the case $p (z) = p^{'} (z) = 0$ we can switch from the polynomial $p (x)$ to the polynomial $p^{'} (x)$ . Note that in this case the coefficients of $p^{'} (x)$ are elements of K as well, the vector of these coefficients is still nonzero, and the degree of $p^{'} (x)$ is smaller than the degree of $p (x)$ . As we can repeat this, we can assume that $p (x) = \sum_{i = 0}^{k} b_{i} \cdot x^{i}$ is a polynomial with a nonzero vector of coefficients $b_{0}, \dots, b_{k} \in K$ , with $p (z) = 0$ (note that this implies $k > 0$ ), and with $p^{'} (z) \neq 0$ . The function $F : R^{k + 2} \to R$ defined by $\begin{array}{r} F (x) : = \sum_{i = 0}^{k} x_{i + 1} \cdot x_{k + 2}^{i}, \end{array}$ for $x = (x_{1}, \dots, x_{k + 2}) \in R^{k + 2}$ is computable and continuously differentiable and has computable partial derivatives. It satisfies $F (b_{0}, \dots, b_{k}, z) = 0$ and $\frac{\partial F}{\partial x_{k + 2}} (b_{0}, \dots, b_{k}, z) \neq 0$ . By the effective version [16, Theorem 4.3] of the Implicit Function Theorem due to McNicholl (and by the fact that, given computable input, a computable functional produces computable output) there exist an open subset $V \subseteq R^{k + 1}$ and a computable and continuously differentiable function $g : V \to R$ such that $(b_{0}, \dots, b_{k}) \in V$ , $g (b_{0}, \dots, b_{k}) = z$ , and, for all $x = (x_{1}, \dots, x_{k + 1}) \in V$ , $\sum_{i = 0}^{k} x_{i + 1} \cdot g {(x)}^{i} = 0$ . As any continuously differentiable function is locally Lipschitz continuous there exists an open subset $W \subseteq V$ with $(b_{0}, \dots, b_{k}) \in W$ such that the restriction $g |_{W}$ is Lipschitz continuous. This restriction $g |_{W}$ is computable as well. Hence, the number z is an element of K. □
Corollary 6.7.
The set $NC$ is a real closed field.
Proof.
This follows from Lemma 6.6 and Proposition 6.5 and the fact that $1 \in NC$ . □

We have seen in Proposition 6.5 that the set $NC$ is closed under computable real functions with open domain. Is it perhaps even closed under arbitary computable real functions? A negative answer is given by the following theorem and by the fact that $NC ∖ EC$ is not empty; see Theorem 7.3 in the following section.
Theorem 6.8.
There exists a computable (and increasing) function $f : R ∖ Q \to R$ such that for every noncomputable real number α the number $f (α)$ is not nearly computable.
Corollary 6.9.
The set $NC$ of nearly computable numbers is not closed under arbitrary computable functions $f : \subseteq R \to R$ .
Proof.
A function f as in Theorem 6.8 maps any element of $NC ∖ EC$ to $R ∖ NC$ . As it is known that $NC ∖ EC$ is not empty (see Theorem 7.3), we conclude that $NC$ is not closed under f. □

In the proof of Theorem 6.8 we use the following proposition. Proposition 6.10.
For a set $A \subseteq N$ the following conditions are equivalent.
A is decidable.

The number $4^{- A} : = \sum_{n \in A} 4^{- n}$ is computable.

The number $4^{- A}$ is nearly computable.

Proof.
The implication “1 ⇒ 2” is obvious. The implication “2 ⇒ 3” follows from Corollary 3.5 and Lemma 5.2. We show “3 ⇒ 1”. Let us consider a set $A \subseteq N$ such that the number $4^{- A}$ is nearly computable. We can assume without loss of generality $A \neq \emptyset$ , hence, $4^{- A} > 0$ . Let ${(a_{n})}_{n}$ be a computable sequence of positive rational numbers that is nearly computably Cauchy and converges to $4^{- A}$ . We define a function $r : N \to N$ by $r (- 1) : = - 1$ and $\begin{array}{r} r (n) : = min {k \in N : k > r (n - 1) and (\exists B \subseteq {0, \dots, n}) | 4^{- B} - a_{k} | < \frac{1}{2} \cdot 4^{- n}} . \end{array}$ The function r is well-defined because the sequence ${(a_{k})}_{k}$ converges to $4^{- A}$ , and we see $\begin{array}{r} | 4^{- (A \cap {0, \dots, n})} - 4^{- A} | ⩽ \sum_{i = n + 1}^{\infty} 4^{- i} = \frac{1}{3} \cdot 4^{- n} < \frac{1}{2} \cdot 4^{- n} . \end{array}$ It is clear that r is increasing and computable. Actually, for every $n \in N$ there is exactly one set $B \subseteq {0, \dots, n}$ with $| 4^{- B} - a_{r (n)} | < \frac{1}{2} \cdot 4^{- n}$ because the numbers $4^{- C}$ for $C \subseteq {0, \dots, n}$ have distance at least $4^{- n}$ from each other. Let us call this set $B_{n}$ . Note that given n we can compute $B_{n}$ . As ${(a_{n})}_{n}$ is nearly computably Cauchy the sequence ${(a_{r (n + 1)} - a_{r (n)})}_{n}$ converges computably to 0. Let $g : N \to N$ be an increasing, computable modulus of convergence of this sequence. We claim that $\begin{aligned} (2) & (\forall m ⩾ g (2 n + 3)) B_{m} \cap {0, 1, \dots, n} = A \cap {0, 1, \dots, n} . \end{aligned}$ Once this is shown it is clear that A is decidable because given n we can compute $B_{g (2 n + 3)}$ and can check whether $n \in B_{g (2 n + 3)}$ or not. Let us show the claim (2). For every $m ⩾ n$ the set $B_{m} \cap {0, 1, \dots, n}$ satisfies $\begin{aligned} | 4^{- (B_{m} \cap {0, 1, \dots, n})} - a_{r (m)} | & ⩽ | 4^{- (B_{m} \cap {0, 1, \dots, n})} - 4^{- B_{m}} | + | 4^{- B_{m}} - a_{r (m)} | \\ < \sum_{i = n + 1}^{m} 4^{- i} + \frac{1}{2} \cdot 4^{- m} = \frac{1}{3} \cdot 4^{- n} - \frac{1}{3} \cdot 4^{- m} + \frac{1}{2} \cdot 4^{- m} \\ = \frac{1}{3} \cdot 4^{- n} + \frac{1}{6} \cdot 4^{- m} \\ ⩽ \frac{1}{2} \cdot 4^{- n}, \end{aligned}$ and it is the only set $C \subseteq {0, 1, \dots, n}$ with $| 4^{- C} - a_{r (m)} | < \frac{1}{2} \cdot 4^{- n}$ because the numbers $4^{- C}$ for $C \subseteq {0, \dots, n}$ have distance at least $4^{- n}$ from each other. By reusing part of the calculation above we obtain for $m ⩾ g (2 n + 3)$ $\begin{aligned} | 4^{- (B_{m + 1} \cap {0, 1, \dots, n})} - a_{r (m)} | & ⩽ | 4^{- (B_{m + 1} \cap {0, 1, \dots, n})} - a_{r (m + 1)} | + | a_{r (m + 1)} - a_{r (m)} | \\ < \frac{1}{3} \cdot 4^{- n} + \frac{1}{6} \cdot 4^{- (m + 1)} + 2^{- (2 n + 3)} \\ = 4^{- n} \cdot (\frac{1}{3} + \frac{1}{6} \cdot 4^{- (m + 1 - n)} + \frac{1}{8}) \\ < \frac{1}{2} \cdot 4^{- n} \end{aligned}$ (for the last step note that $m ⩾ g (2 n + 3) ⩾ 2 n + 3$ because g is assumed to be increasing). So, $B_{m + 1} \cap {0, 1, \dots, n}$ is also a set $C \subseteq {0, 1, \dots, n}$ with $| 4^{- C} - a_{r (m)} | < \frac{1}{2} \cdot 4^{- n}$ . But we just saw that there can be only one set C with this property, and that is the set $B_{m} \cap {0, 1, \dots, n}$ . We conclude $\begin{aligned} (3) & (\forall m ⩾ g (2 n + 3)) B_{m} \cap {0, 1, \dots, n} = B_{m + 1} \cap {0, 1, \dots, n} . \end{aligned}$ For m so large that $| 4^{- A} - a_{r (m)} | < \frac{1}{6} \cdot 4^{- n}$ we similarly observe that $\begin{array}{r} | 4^{- (A \cap {0, 1, \dots, n})} - a_{r (m)} | ⩽ | 4^{- (A \cap {0, 1, \dots, n})} - 4^{- A} | + | 4^{- A} - a_{r (m)} | < \sum_{i = n + 1}^{\infty} 4^{- i} + \frac{1}{6} \cdot 4^{- n} = \frac{1}{2} \cdot 4^{- n} . \end{array}$ By the same uniqueness argument as above we conclude that for sufficiently large m we have $B_{m} \cap {0, 1, \dots, n} = A \cap {0, 1, \dots, n}$ . This and (3) imply (2). This ends the proof of the implication “3 ⇒ 1”. □
Proof of Theorem 6.8.
We define a function $f : R ∖ Q \to R$ by $\begin{array}{r} f (α) : = \sum_{n : ν_{Q} (n) < α} 4^{- n}, \end{array}$ for $α \in R ∖ Q$ . This function is computable because, for any n and any irrational number α, if one knows α with sufficient precision then one can decide whether $ν_{Q} (n) < α$ or not. It is clear that the function f is increasing.

Let us consider some irrational real number α such that $f (α)$ is nearly computable. Then by Proposition 6.10 the set ${n \in N : ν_{Q} (n) < α}$ is decidable. Hence, α is computable. □

7. Left computable, nearly computable numbers

In this section we are going to have a look at the set $LC \cap NC$ of real numbers that are at the same time left-computable and nearly computable.

A subset $A \subseteq {0, 1}^{*}$ is called prefix-free if, for all $u, v \in A$ , if $u \subseteq v$ then $u = v$ , in other words, if no string u in A is a proper prefix of any other string v in A. We shall make use of the following well-known effective version of the Kraft theorem, sometimes called Kraft–Chaitin Theorem; see, e.g., [4, Theorem 3.6.1].

Lemma 7.1.
Let $f : N \to N$ be a computable function such that $\sum_{n = 0}^{\infty} 2^{- f (n)} ⩽ 1$ . Then there exists an injective computable function $h : N \to {0, 1}^{}$ such that the set* ${h (n) : n \in N}$ is prefix-free and $| h (n) | = f (n)$ , for all $n \in N$ .

The equivalence of the conditions (4) and (6) in the following proposition is essentially the content of Theorem 5 in [24].
Proposition 7.2.
For a left-computable number α the following four conditions are equivalent:
It is nearly computable.

There exists an increasing, computable, and nearly computably Cauchy sequence ${(a_{n})}_{n}$ of rational numbers converging to α.

Every nondecreasing, computable sequence ${(a_{n})}_{n}$ of real numbers converging to α is nearly computably Cauchy.

For every increasing, computable sequence ${(a_{n})}_{n}$ of rational numbers converging to α the sequence ${(a_{n + 1} - a_{n})}_{n}$ converges computably to 0.
For a left-computable number α with $α ⩾ 0$ the following condition is equivalent to these conditions as well:
Every computable sequence ${(b_{n})}_{n}$ of positive rational numbers with $\sum_{i = 0}^{\infty} b_{i} = α$ converges computably to 0.
For a left-computable number $α \in [0, 1]$ the following condition is equivalent to these conditions as well:
Every prefix-free, computably enumerable set $A \subseteq {0, 1}^{}$ with* $α = \sum_{w \in A} 2^{- | w |}$ is decidable.

Proof of Proposition 7.2.
“(1) ⇒ (2)”: Let ${(a_{n})}_{n}$ be a computable and nearly computably Cauchy sequence of rational numbers converging to α. Let ${(b_{n})}_{n}$ be a computable and increasing sequence of rational numbers converging to α. As both sequences have the same limit and are computable there exists an increasing, computable function $s : N \to N$ such that, for all n, $| a_{s (n)} - b_{s (n)} | ⩽ 2^{- n}$ . By Lemma 5.9 the sequence ${(a_{s (n)})}_{n}$ is nearly computably Cauchy as well, and it is clear that it converges to α as well. By Lemma 5.8 the sequence ${(b_{s (n)})}_{n}$ is nearly computably Cauchy as well, and it is clear that it converges to α as well. Furthermore, this sequence is increasing.

“(2) ⇒ (1)”: Trivial.

“(2) ⇔ (3)”: By Proposition 5.11.

“(3) ⇒ (4)”: Trivial.

“(4) ⇒ (2)”: Let ${(a_{n})}_{n}$ be an increasing, computable sequence in $Q$ converging to α. Let $r : N \to N$ be a total, increasing, computable function. We define $b_{n} : = a_{r (n)}$ . Then the sequence ${(b_{n})}_{n}$ is an increasing, computable sequence in $Q$ converging to α as well. By (4) the sequence ${(a_{r (n + 1)} - a_{r (n)})}_{n} = {(b_{n + 1} - b_{n})}_{n}$ converges computably to 0. Hence, ${(a_{n})}_{n}$ is nearly computably Cauchy.

In the rest of the proof we assume that $α ⩾ 0$ .

“(4) ⇒ (5)”: If $α = 0$ then there is no sequence ${(b_{n})}_{n}$ of positive rational numbers with $\sum_{i = 0}^{\infty} b_{i} = α$ . Hence, in this case nothing needs to be shown. Let us consider the case $α > 0$ . Let ${(b_{n})}_{n}$ be a computable sequence of positive rational numbers with $\sum_{i = 0}^{\infty} b_{i} = α$ . We define $a_{n} : = \sum_{i = 0}^{n - 1} b_{i}$ , for all n. Then the sequence ${(a_{n})}_{n}$ is an increasing, computable sequence of rational numbers converging to α. By (4) the sequence ${(a_{n + 1} - a_{n})}_{n} = {(b_{n})}_{n}$ converges computably to 0.

“(5) ⇒ (4)”: Let us first consider the case $α = 0$ . Then α is a computable number. Let us consider an increasing, computable sequence ${(a_{n})}_{n}$ of rational numbers converging to 0. By Lemma 4.6 it converges computably, by Corollary 3.5 it is computably Cauchy, by Lemma 5.2 it is nearly computably Cauchy. Hence, in particular the sequence ${(a_{n + 1} - a_{n})}_{n}$ converges computably to 0. Now let us consider the case $α > 0$ . Let us consider an increasing, computable sequence ${(a_{n})}_{n}$ of rational numbers converging to α. There exists some $N > 0$ with $a_{N} > 0$ . Let the sequence ${(c_{n})}_{n}$ be defined by $c_{0} : = 0$ and $c_{n} : = a_{N + n}$ , for $n > 0$ . Then the sequence ${(c_{n})}_{n}$ is an increasing computable sequence of rational numbers converging to α. Let the sequence ${(b_{n})}_{n}$ be defined by $b_{n} : = c_{n + 1} - c_{n}$ . Then ${(b_{n})}_{n}$ is a computable sequence of positive rational numbers with $\sum_{i = 0}^{\infty} b_{i} = α$ . By (5) it converges computably to 0. That means, the sequence ${(c_{n + 1} - c_{n})}_{n}$ converges computably to 0. As the sequences ${(c_{n + 1} - c_{n})}_{n}$ and ${(a_{n + 1} - a_{n})}_{n}$ are identical up to a finite prefix, the sequence ${(a_{n + 1} - a_{n})}_{n}$ converges computably to 0 as well.

In the rest of the proof we assume that $α \in [0, 1]$ .

“(5) ⇒ (6)”: Let $A \subseteq {0, 1}^{}$ be a prefix-free, computably enumerable set with $α = \sum_{w \in A} 2^{- | w |}$ . If A is finite then it is decidable. So, let us assume that A is infinite. Let $h : N \to {0, 1}^{}$ be a total, injective, computable function with $range (h) = A$ . We define $b_{n} : = 2^{- | h (n) |}$ , for all $n \in N$ . Then ${(b_{n})}_{n}$ is a computable sequence of positive rational numbers with $\sum_{i = 0}^{\infty} b_{i} = α$ . By (5) the sequence ${(b_{n})}_{n}$ converges computably. It is clear that it converges to 0. Hence, there exists a total, computable function $g : N \to N$ such that, for all $m, n \in N$ , $\begin{array}{r} m ⩾ g (n) \Rightarrow | b_{m} | ⩽ 2^{- n} . \end{array}$ Hence, for all $m, n \in N$ , $\begin{array}{r} m ⩾ g (n) \Rightarrow | h (m) | ⩾ n . \end{array}$ But this shows that A is decidable: A string w of length n is an element of A if, and only if, it is an element of ${h (0), \dots, h (g (n + 1) - 1)}$ .

“(6) ⇒ (5)”: In the case $α = 0$ nothing needs to be shown. So, we consider the case $0 < α ⩽ 1$ . We assume that (6) is true. Let ${(b_{n})}_{n}$ be a computable sequence of positive rational numbers with $\sum_{i = 0}^{\infty} b_{i} = α$ . We wish to show that it converges computably. We define a function $f : N \to N$ recursively as follows: $\begin{array}{r} f (0) : = min {k \in N : 2^{- k} < b_{0}}, \end{array}$ and, assuming that we have defined $f (0), \dots, f (n)$ so that $\sum_{i = 0}^{n} 2^{- f (i)} < \sum_{i = 0}^{n} b_{i}$ , we define $\begin{array}{r} f (n + 1) : = min {k \in N : \sum_{i = 0}^{n} 2^{- f (i)} + 2^{- k} < \sum_{i = 0}^{n + 1} b_{i}} . \end{array}$ Then f is total and computable and satisfies $\sum_{i = 0}^{\infty} 2^{- f (i)} = α$ . Furthermore, $2^{- f (n)} ⩾ b_{n} / 2$ , for all n. By Lemma 7.1 there exists a total, injective, computable function $h : N \to {0, 1}^{}$ such that the set $A : = h (N) = {h (n) : n \in N}$ is prefix-free and satisfies $| h (i) | = f (i)$ , for all $i \in N$ . The set A is computably enumerable. By (6) it is decidable. The function $g : N \to N$ defined by $\begin{array}{r} g (n) : = min {k \in N : A \cap (⋃_{i = 0}^{n} {0, 1}^{i}) \subseteq {h (0), \dots, h (k)}}, \end{array}$ for all $n \in N$ is total and computable. We observe that, for all $i > g (n)$ , $\begin{array}{r} f (i) = | h (i) | ⩾ n + 1, \end{array}$ hence, $b_{i} / 2 ⩽ 2^{- f (i)} ⩽ 2^{- (n + 1)}$ . This shows that the sequence ${(b_{n})}_{n}$ converges computably. □

The main insight in the proof of the following theorem is due to Downey and LaForte [7].
Theorem 7.3.
$EC ⊊ LC \cap NC$ .
Proof.
The inclusion $EC \subseteq LC$ is clear. The inclusion $EC \subseteq NC$ was already stated in Proposition 6.2. The fact that there exists a real number that is left-computable and nearly computable but not computable follows from Proposition 7.2 and from the main result by Downey and LaForte [7] who proved that there exists a real number in the interval $[0, 1]$ that is left-computable but not computable and satisfies Condition (6) in Proposition 7.2. □

The proof of the main result of Downey and LaForte [7] is by an infinite injury priority argument. Thus, one may say that the set of real numbers that are left-computable and nearly computably is only slightly larger than the set of computable numbers.

An interesting property of the set of left-computable nearly computable real numbers is that it is closed under Solovay reducibility [23]. A left-computable number α is Solovay reducible* to a left-computable number β (in symbols: $α ⩽_{S} β$ ) if there exist a computable function $f : \subseteq Q \to Q$ and a positive rational number D such that ${q \in Q | q < β} \subseteq dom (f)$ and, for all $q \in Q$ with $q < β$ , we have $f (q) < α$ and $α - f (q) < D \cdot (β - q)$ . We write $α <_{S} β$ iff $α ⩽_{S} β$ and $β ≰_{S} α$ . We shall use the following two characterisations of Solovay reducibility.
Lemma 7.4 (Downey, Hirschfeldt, and Nies [6]).

Let α and β be two left-computable real numbers. Then the following conditions are equivalent.

α is Solovay reducible to β.

There exist a positive rational number D and two computable, increasing sequences ${(a_{n})}_{n}$ and ${(b_{n})}_{n}$ of rational numbers with ${lim}_{n \to \infty} a_{n} = α$ , ${lim}_{n \to \infty} = β$ and, for all n, $a_{n + 1} - a_{n} ⩽ D \cdot (b_{n + 1} - b_{n})$ .

There exists a positive rational number D such that the number $D \cdot β - α$ is left-computable.

Proposition 7.5.
Let α and β be two left-computable real numbers. If $α ⩽_{S} β$ and β is nearly computable then α is nearly computable as well.
Proof.
According to Lemma 7.4, there exist a rational number $D > 0$ and computable, increasing sequences ${(a_{n})}_{n}$ and ${(b_{n})}_{n}$ of rational numbers with ${lim}_{n \to \infty} a_{n} = α$ , ${lim}_{n \to \infty} = β$ and $a_{n + 1} - a_{n} ⩽ D \cdot (b_{n + 1} - b_{n})$ for all n. According to Proposition 7.2.2 it is sufficient to show that the sequence ${(a_{n})}_{n}$ is nearly computably Cauchy. Let us consider a computable and increasing function $r : N \to N$ . By Proposition 7.2.3 the sequence ${(b_{n})}_{n}$ is nearly computably Cauchy. Thus, the sequence ${(b_{r (n + 1)} - b_{r (n)})}_{n}$ converges computably to 0. As for all $n \in N$ we have $a_{r (n + 1)} - a_{r (n)} ⩽ D \cdot (b_{r (n + 1)} - b_{r (n)})$ , the sequence ${(a_{r (n + 1)} - a_{r (n)})}_{n}$ converges computably to 0 as well. Hence, the sequence ${(a_{n})}_{n}$ is nearly computably Cauchy. □

As a corollary, we obtain nearly computable real numbers that are neither left-computable nor right-computable (a real number x is called right-computable if $- x$ is left-computable), which answers a question posed by a referee of an earlier version of this paper.
Corollary 7.6.
There exist nearly computable numbers that are neither left-computable nor right-computable.
Proof.
Let γ be a left-computable, nearly computable number. By [24, Theorem 6] γ is not Martin-Löf random (see also Corollary 8.4). By [6, Theorem 3.1] there exist left-computable numbers $α <_{S} γ$ and $β <_{S} γ$ with $α + β = γ$ . It is well known and easy to see that this implies that they are Solovay incomparable, that is, satisfy $α ≰_{S} β$ and $β ≰_{S} α$ . So, according to Lemma 7.4 the number $δ : = α - β$ is neither left-computable nor right-computable. But, according to Proposition 7.5, the numbers α and β are nearly computable, and, according to Proposition 6.5 or Corollary 6.7, the number δ is nearly computable as well. □

Often the requirement that a real number should be left-computable and nearly computable in combination with some other effectivity requirement forces the real number to be even computable. In this section we discuss two observations of this kind and in Section 8 three further ones.
Definition 7.7.
For a set $A \subseteq N$ let $0 . A : = \sum_{n \in A} 2^{- n - 1}$ .

If one identifies a set $A \subseteq N$ with its characteristic function $χ_{A} \in {0, 1}^{N}$ , an infinite binary sequence, then one can say that $0 . A$ is the real number in the interval $[0, 1]$ with binary representation $0 . A$ . Note that the function $A \mapsto 0 . A$ is a surjection from the power set $P (N) : = {A : A \subseteq N}$ onto the real interval $[0, 1]$ . It is also well known and easy to see that for any set $A \subseteq N$ the real number $0 . A$ is computable if, and only if, A is decidable.
Definition 7.8.
A real number α is called strongly left-computable if there exists a computably enumerable set $A \subseteq N$ such that $α = 0 . A$ .

Any computable number in $[0, 1]$ is strongly left-computable. The existence of a c.e. set $A \subseteq N$ that is not decidable implies that there are strongly left-computable real numbers that are not computable. It is clear that any strongly left-computable number is left-computable. But it is also well known and easy to see that there are left-computable numbers in the interval $[0, 1]$ that are not strongly left-computable; see Soare [21]. So, any computable number in $[0, 1]$ is strongly left-computable and nearly computable. An argument by Downey, Hirschfeldt, and LaForte [5, Pages 105, 106] shows that the converse is true as well.
Proposition 7.9 (Downey, Hirschfeldt, and LaForte [5, Pages 105, 106]).

Any strongly left-computable and nearly computable number is computable.

For completeness sake we give the easy proof.

Proof.
Let us consider a computably enumerable set $A \subseteq N$ such that the real number $0 . A$ is nearly computable. We claim that $0 . A$ is even computable. It is sufficient to show that A is decidable. If A is finite this is clear anyway. So, let us assume that A is infinite. Let $f : N \to N$ be a total, injective, computable function with $f (N) = A$ . Then the sequence ${(b_{n})}_{n}$ defined by $b_{n} : = 2^{- f (n)}$ , for $n \in N$ , is a computable sequence of positive rational numbers with $\sum_{i = 0}^{\infty} b_{i} = 2^{- A}$ . As we assume that $0 . A$ is nearly computable, by Proposition 7.2 the sequence ${(b_{n})}_{n}$ converges computably to 0. Let $g : N \to N$ be a computable modulus of convergence of this sequence. Remember that this implies that for $m, n \in N$ with $m ⩾ g (n)$ we have $b_{m} ⩽ 2^{- n}$ , hence, $f (m) ⩾ n$ . Hence, a number n is an element of A if, and only if, n is an element of ${f (0), \dots, f (g (n + 1) - 1)}$ . This shows that A is decidable. □

Remember that a real number α is called computably approximable if there exists a computable sequence ${(q_{n})}_{n}$ of rational numbers that converges to α, and that by $CA$ we denote the set of all computably approximable numbers.
Corollary 7.10.

$LC ⊈ NC$ and $NC ⊈ LC$ .

$NC ⊊ CA$ .

Proof.
Let $K \subseteq N$ be a computably enumerable set that is not decidable. Then the number $0 . K$ is strongly left-computable, hence also left-computable, but not computable. By Proposition 7.9 it is not nearly computable. This shows $LC ⊈ NC$ .

The inclusion $NC \subseteq CA$ is clear. As any left-computable number is computably approximable, the number $0 . K$ is an element of $CA$ but not of $NC$ . Hence, $NC ⊊ CA$ .

By Theorem 7.3 there exists a real number α that is left-computable and nearly computable but not computable. Then the number $- α$ is not left-computable (otherwise α would even be computable) but nearly computable (this follows from Proposition 5.10). This shows $NC ⊈ LC$ . □

Let us deduce some observations concerning the possible positions of the elements of the following set $\begin{array}{r} Θ : = {A \subseteq N : 0 . A is a nearly computable number} \end{array}$ in the arithmetical hierarchy. Let $Σ_{n}^{0}$ , $Π_{n}^{0}$ , and $Δ_{n}^{0}$ be the usual classes of the arithmetical hierarchy; see, e.g., [22].
Corollary 7.11.

$Σ_{0}^{0} = Π_{0}^{0} = Δ_{0}^{0} = {A \subseteq N : A is decidable} ⊊ Θ$ .

$Σ_{1}^{0} \cap Θ = Π_{1}^{0} \cap Θ = Σ_{0}^{0}$ .

$Θ ⊊ Δ_{2}^{0}$ .

Proof.

On the one hand, remember that for $A \subseteq N$ the number $0 . A$ is computable iff A is decidable and that any computable number is nearly computable. On the other hand, by Theorem 7.3 there exists a real number that is nearly computable but not computable.

The equality $Σ_{1}^{0} \cap Θ = Σ_{0}^{0}$ follows directly from Proposition 7.9. The inclusion $Σ_{0}^{0} \subseteq Π_{1}^{0} \cap Θ$ follows from the first assertion. For the inclusion $Π_{1}^{0} \cap Θ \subseteq Σ_{0}^{0}$ let us consider a set $A \in Π_{1}^{0} \cap Θ$ . Then the set $B : = N ∖ A$ is an element of $Σ_{1}^{0}$ , and the number $0 . B = 1 - 0 . A$ is nearly computable. We conclude $B \in Σ_{1}^{0} \cap Θ = Σ_{0}^{0}$ , hence, $A \in Σ_{0}^{0}$ .

This follows from $NC ⊊ CA$ and from the fact that a set $A \subseteq N$ is an element of $Δ_{2}^{0}$ if, and only if, the number $0 . A$ is computably approximable; see Ho [10].
□

The second observation of the kind described above (that asking a real number to be left-computable and nearly computable and to satisfy some other effectivity condition forces the number to be computable) that we wish to mention is the following result by Wu [26].
Proposition 7.12 (Wu [26]).

There exists an undecidable, c.e. set $B \subseteq N$ such that every left-computable and nearly computable number α with $α ⩽_{T} B$ is computable.

Here $⩽_{T}$ denotes Turing reduction; see Soare [22]. A set $A \subseteq N$ is Turing reducible to a set $B \subseteq N$ , written $A ⩽_{T} B$ , if there exists an oracle Turing machine with oracle B which computes A. By $α ⩽_{T} B$ for a real number α we mean $A ⩽_{T} B$ where $A \subseteq N$ is the unique coinfinite set with $(α - 0 . A) \in Z$ . The proof of Wu’s result is a complicated $\emptyset^{‴}$ -priority argument.

8. Nearly computable numbers, a genericity notion and randomness notions

The third observation of the kind described in the previous section (that asking a real number to be left-computable and nearly computable and to satisfy some other effectivity condition forces the number to be computable) that we wish to mention is the following result by Hoyrup [11].

Proposition 8.1 (Hoyrup [11, Prop. 4.3.2]).

If a real number is left-computable and nearly computable then it is either computable or generic from the right.

Corollary 8.2 (Hoyrup [11]).

If a real number is left-computable and nearly computable then it is either computable or weakly 1-generic.

Note that the assertion in the corollary is equivalent to the following: If a real number is left-computable, nearly computable, and not weakly 1-generic then it is computable.

Let us explain the notions appearing in these assertions. A subset $U \subseteq R$ of the real line is called effectively open if there exists a c.e. set $A \subseteq N$ such that $U = ⋃_{n \in A} B_{n}^{(1)}$ . Hoyrup [11] calls a real number x generic from the right if for every effectively open set $U \subseteq R$ either $x \in U$ or there exists a number $ε > 0$ with $[x, x + ε) \cap U = \emptyset$ . A real number x is called weakly 1-generic if it is an element of every dense effectively open subset $U \subseteq R$ [14]. Clearly, every real number that is generic from the right is weakly 1-generic as well [11, Prop. 4.3.1]. Hence, Corollary 8.2 follows directly from Proposition 8.1. Note that a computable number x cannot be weakly 1-generic because for a computable number x the set $R ∖ {x}$ is a dense and effectively open subset of $R$ . One cannot delete the assumption that the real number should be left-computable from Proposition 8.1 because by Theorem 7.3 and Corollary 6.7 there exist right-computable numbers that are nearly computable but not computable and a right-computable number cannot be generic from the right (because for a right-computable number x the set $(x, \infty)$ is an effectively open set). But one can delete this assumption from Corollary 8.2.

Theorem 8.3.
If a real number is nearly computable then it is either computable or weakly 1-generic.
Proof.
For the sake of a contradiction, let us assume that there exists a real number y that is nearly computable but neither computable nor weakly 1-generic. Let $U \subseteq R$ be a dense, effectively open set with $y \notin U$ . There exists a total computable function $f : N \to N$ such that $U = ⋃_{n \in N} B_{f (n)}^{(1)}$ . Let ${(x_{n})}_{n}$ be a computable sequence of rational numbers that converges to y and is nearly computably Cauchy.

We define an increasing computable function $s : N \to N$ as follows. We set $s (0) : = 0$ . Let us consider a number $n > 0$ and let us assume that we have fixed the values $s (0) < \dots < s (n - 1)$ . Now we search for a number $m > s (n - 1)$ such that $\begin{array}{r} x_{m} \notin ⋃_{i = 0}^{n - 1} {\overline{B}}_{f (i)}, \end{array}$ where ${\overline{B}}_{n}$ is the closure of $B_{n}^{(1)}$ , that is, ${\overline{B}}_{n} = [ν_{Q} π_{1} (n) - 2^{- π_{2} (n)}, ν_{Q} π_{1} (n) - 2^{- π_{2} (n)}]$ , for all $n \in N$ . Such a number m exists because $y \notin U$ and y is not rational, hence, $y \notin ⋃_{i = 0}^{n - 1} {\overline{B}}_{f (i)}$ , and because the sequence ${(x_{k})}_{k}$ converges to y. Then we set $s (n) : = m$ .

The sequence ${(z_{n})}_{n}$ defined by $z_{n} : = x_{s (n)}$ is a computable sequence of rational numbers that converges to y and is nearly computably Cauchy by Lemma 5.9. And it has the property that $\begin{array}{r} z_{n} \notin ⋃_{i = 0}^{n - 1} {\overline{B}}_{f (i)}, \end{array}$ for all $n \in N$ . Let $g : N \to N$ be a computable modulus of convergence of the sequence ${(| z_{n + 1} - z_{n} |)}_{n}$ . Then, for all $m, n$ with $m ⩾ g (n)$ , we have $| z_{m + 1} - z_{m} | ⩽ 2^{- n}$ . We define a computable function $label : N \to {left, right}$ by $r (n) : = max {n + 1, g (π_{2} (f (n)))}$ and $\begin{array}{r} label (n) : = \{\begin{array}{ll} left & if ν_{Q} (π_{1} (f (n))) < z_{r (n)}, \\ right & if z_{r (n)} < ν_{Q} (π_{1} (f (n))), \end{array} \end{array}$ for $n \in N$ (note that $ν_{Q} (π_{1} (f (n)))$ is the midpoint of the closed interval ${\overline{B}}_{f (n)}$ and $r (n) > n$ , hence, $z_{r (n)} \notin {\overline{B}}_{f (n)}$ , for all n). Now note that, for all $m ⩾ r (n)$ , on the one hand $z_{m} \notin {\overline{B}}_{f (n)}$ , on the other hand, $| z_{m + 1} - z_{m} | ⩽ 2^{- π_{2} (f (n))}$ and the length of the interval ${\overline{B}}_{f (n)}$ is equal to $2 \cdot 2^{- π_{2} (f (n))}$ . Hence, for all n with $label (n) = left$ and for all $m ⩾ r (n)$ the interval ${\overline{B}}_{f (n)}$ is to the left of $z_{m}$ . We conclude that for all n with $label (n) = left$ this interval is to the left of y. Similarly, for all n with $label (n) = right$ the interval ${\overline{B}}_{f (n)}$ is to the right of y. As the set U is a dense subset of $R$ , we can use the function $label$ in order to compute y with arbitrary precision, hence, y is computable. Contradiction. □

For the notions “hyperimmune” and “Martin-Löf random”, that appear in the following corollary, the reader is referred to [4,17].
Corollary 8.4.

If a real number is nearly computable then it is either computable or hyperimmune.

If a real number is nearly computable then it is not Martin-Löf random.

Note that the first assertion is equivalent to the following: If a real number is nearly computable and not hyperimmune then it is computable.
Proof.
The first assertion follows from Theorem 8.3 and from the following result by Kurtz [14, Corollary 2.7]: every weakly 1-generic real is hyperimmune (for a short proof of this fact see [17, Prop. 1.8.48]). The second assertion follows from the first because a Martin-Löf random real is neither computable nor hyperimmune; see, e.g., [24]. □

This corollary strengthens Theorem 6 by Stephan and Wu [24, Theorem 6], who had shown the same assertions under the additional assumption that the real number is left-computable. This was the fourth observation of the kind described in the previous section (that asking a real number to be left-computable and nearly computable and to satisfy some other effectivity condition forces the number to be computable) that we wanted to mention.

In order to state the next results we need to introduce some further notions. Let $Σ : = {0, 1}$ be the binary alphabet. For any function $f : \subseteq Σ^{} \to Σ^{}$ defined on a subset of $Σ^{}$ and with values in $Σ^{}$ the Kolmogorov complexity $K_{f} (τ)$ of a binary string τ is $\begin{array}{r} K_{f} (τ) : = \{\begin{array}{ll} \infty & if there is no σ \in dom (f) with f (σ) = τ, \\ min {| σ | : σ \in dom (f), f (σ) = τ} & otherwise . \end{array} \end{array}$ A computable function $f : \subseteq Σ^{} \to Σ^{}$ is called universal if its domain $dom (f)$ is a prefix-free subset of $Σ^{}$ and if for every computable function $g : \subseteq Σ^{} \to Σ^{}$ with prefix-free domain there exists a constant $c \in N$ such that for any binary string τ with $K_{g} (τ) < \infty$ we have $K_{f} (τ) ⩽ K_{g} (τ) + c$ . It is well known that there exists a universal function $f : \subseteq Σ^{} \to Σ^{}$ ; see, e.g., [4,17]. Note that if $f : \subseteq Σ^{} \to Σ^{}$ is a universal function then $K_{f} (τ) < \infty$ , for all $τ \in Σ^{}$ . In the rest of this section we fix a universal function $U : \subseteq Σ^{} \to Σ^{}$ and write K instead of $K_{U}$ .

For any finite binary string $w \in Σ^{}$ let $0 . w$ be the rational number in $[0, 1]$ with the finite binary expansion w, that is, $0 . w : = \sum_{n = 0}^{| w | - 1} 2^{- n - 1} \cdot w (n)$ . Similarly, for any $α \in Σ^{N}$ let $0 . α$ be the real number in $[0, 1]$ defined by $0 . α : = \sum_{n = 0}^{\infty} 2^{- n - 1} \cdot α (n)$ . In this case α is called a binary representation* of the real number $0 . α$ . Note that only the real numbers in $[0, 1]$ of the form $z / 2^{k}$ for $k \in N$ and $z \in Z$ with $0 ⩽ z ⩽ 2^{k}$ have two binary representations. All other real numbers in $[0, 1]$ have exactly one binary representation.
Definition 8.5.

An infinite binary sequence α is called K-trivial if there exists a constant $c \in N$ such that, for all $n \in N$ , $K (α ↾ n) ⩽ K (0^{n}) + c$ , where $0^{n}$ is the string of length n consisting only of zeros.

An infinite binary sequence α is called infinitely often K-trivial if there exists a constant $c \in N$ such that, for infinitely many $n \in N$ , $K (α ↾ n) ⩽ K (0^{n}) + c$ .

Following [24] we call an infinite binary sequence α strongly Kurtz random if there does not exist a total computable function $f : N \to N$ such that, for all $n \in N$ , $K (α ↾ f (n)) < f (n) - n$ .

A real number $x \in [0, 1]$ is called K-trivial respectively infinitely often K-trivial respectively strongly Kurtz random if at least one (and then all) of its binary representations is (are) K-trivial respectively strongly Kurtz random.
Corollary 8.6.
Every nearly computable number in $[0, 1]$ is infinitely often K-trivial.
Proof.
This follows from Theorem 8.3, from the fact that every computable real number in $[0, 1]$ is K-trivial, hence, infinitely often K-trivial, and from the following result by Barmpalias and Vlek [3, Theorem 2.9]: every weakly 1-generic real in $[0, 1]$ is infinitely often K-trivial. □

This raises the question whether a nearly computable number can or must be K-trivial. For nearly computable numbers that are left-computable this question was answered by Stephan and Wu [24]. The following result by them is the fifth observation of the kind described in the previous section (that asking a real number to be left-computable and nearly computable and to satisfy some other effectivity condition forces the number to be computable) that we wish to mention.
Proposition 8.7 (Stephan and Wu [24, Theorems 3 and 4]).

If a real number in $[0, 1]$ is left-computable and nearly computable then it is either computable or strongly Kurtz random.

If a real number in $[0, 1]$ is left-computable, nearly computable, and K-trivial then it is computable.

Note that the first assertion is equivalent to the following: If a real number is left-computable, nearly computable, and not strongly Kurtz random then it is computable.

We are going to show that, similarly as above, this proposition can be strengthened as follows: One can omit the assumption that the real number is left-computable.

Theorem 8.8.

If a real number in $[0, 1]$ is nearly computable then it is either computable or strongly Kurtz random.

If a real number in $[0, 1]$ is nearly computable and K-trivial then it is computable.

Proof.
The second assertion follows from the first assertion because a K-trivial binary sequence cannot be strongly Kurtz random. For completeness sake we prove this, similarly to the argument in [24, Page 468]. Let $α \in Σ^{N}$ be K-trivial. Then there exists a constant $c \in N$ such that, for all $n \in N$ , $K (α ↾ n) ⩽ K (0^{n}) + c$ . And it is well known that there exists a constant $d \in N$ such that, for all $n \in N$ , $K (0^{n}) < \frac{n}{2} + d$ . The function $f : N \to N$ defined by $f (n) : = 2 \cdot (n + c + d)$ is computable and satisfies, for all $n \in N$ , $\begin{array}{r} K (α ↾ f (n)) ⩽ K (0^{f (n)}) + c < \frac{f (n)}{2} + d + c = f (n) - n . \end{array}$ Hence, α is not strongly Kurtz random.

We prove the first assertion. Let $α \in Σ^{N}$ be an infinite binary sequence that is not strongly Kurtz random and such that the real number $0 . α$ is nearly computable. We wish to show that the real number $0 . α$ is computable. This is clear if α contains only finitely many zeros or only finitely many ones. So, in the following we assume that α contains infinitely many zeros and infinitely many ones. Let $f : N \to N$ be a total computable function such that $K (α ↾ f (n)) < f (n) - n$ , for all n. This implies $f (n) > n$ , for all n. Let ${(a_{n})}_{n}$ be a computable sequence of rational numbers in $[0, 1]$ that is nearly computably Cauchy and converges to $0 . α$ . For each n we can compute a finite binary string $u_{n}$ with $| 0 . u_{n} - a_{n} | < 2^{- n}$ . Then the sequence ${(u_{n})}_{n}$ of binary strings is computable, and the sequence ${(0 . u_{n})}_{n}$ of rational numbers is nearly computably Cauchy as well (by Lemma 5.8) and converges to $0 . α$ as well. For each $n \in N$ let $β_{n} \in Σ^{N}$ be the infinite binary sequence obtained by appending to $u_{n}$ infinitely many zeros, that is, $β_{n} : = u_{n} 0^{N}$ . As α contains infinitely many zeros and infinitely many ones, the sequence ${(β_{n})}_{n}$ converges to α (in the sense that for every $ℓ \in N$ there exists an $m \in N$ such that, for all $n ⩾ m$ , $β_{n} ↾ ℓ = α ↾ ℓ$ ). We define an increasing and computable function $s : N \to N$ as follows. In order to define $s (0)$ we search for a number t such that $K (β_{t} ↾ f (0)) < f (0)$ (the search is done using a fixed computable enumeration of the domain $dom (U)$ of the universal function U). Once we have found such a t we set $s (0) : = t$ . Note that $K (β_{s (0)} ↾ f (0)) < f (0) - 0$ . Let us assume that we have defined $s (0), \dots, s (n)$ so that
$s (0) < s (1) < \dots < s (n)$ ,

for all $k, m$ with $0 ⩽ k ⩽ m ⩽ n$ , $K (β_{s (m)} ↾ f (k)) < f (k) - k$ .
In order to define $s (n + 1)$ we search for a number $t > s (n)$ such that, for all $k ⩽ n + 1$ , $K (β_{t} ↾ f (k)) < f (k) - k$ . Once we have found such a t we set $s (n + 1) : = t$ . As, by assumption $K (α ↾ f (n)) < f (n) - n$ , for all n, in this way an increasing, computable function $s : N \to N$ is defined such that, $\begin{array}{r} for all m, n, if m ⩾ n then K (β_{s (m)} ↾ f (n)) < f (n) - n . \end{array}$

The sequence ${(0 . u_{s (n)})}_{n}$ is a computable sequence of rational numbers. As the sequence ${(0 . u_{n})}_{n}$ is nearly computably Cauchy, the sequence ${(0 . u_{s (n + 1)} - 0 . u_{s (n)})}_{n}$ converges computably to 0. Let $g : N \to N$ be a computable and increasing modulus of convergence of this sequence. We claim that, for all n, $\begin{aligned} (4) & | 0 . u_{s (g (f (n)))} - 0 . α | ⩽ 2^{- n} . \end{aligned}$ As the sequence of rational numbers ${(0 . u_{s (g (f (n)))})}_{n}$ is computable, this claim shows that $0 . α$ is computable. Let us prove the claim (4). Let us consider some $n \in N$ . Let $\begin{array}{r} U_{n} : = {w \in Σ^{f (n)} : min {0 . α, 0 . u_{s (g (f (n)))}} < 0 . w < max {0 . α, 0 . u_{s (g (f (n)))}}} . \end{array}$ Let $\begin{array}{r} V_{n} : = {w \in Σ^{f (n)} : (\exists m ⩾ g (f (n))) w = β_{s (m)} ↾ f (n)} . \end{array}$ We make the following three observations.
$| 0 . u_{s (g (f (n)))} - 0 . α | ⩽ (| U_{n} | + 1) \cdot 2^{- f (n)}$ (here $| U_{n} |$ is the number of elements of the finite set $U_{n}$ ). This is an immediate consequence of the definition of $U_{n}$ .

$U_{n} \subseteq V_{n}$ . This is due to the fact that, for all $m ⩾ g (f (n))$ , $| 0 . u_{s (m + 1)} - 0 . u_{s (m)} | ⩽ 2^{- f (n)}$ . The rational numbers $0 . u_{s (m)}$ for $m ⩾ g (f (n))$ converge to α, and they do this in small steps so that all intervals of the form $[0 . w, 0 . w + 2^{- f (n)})$ for $w \in U_{n}$ must be visited by them.

$| V_{n} | ⩽ 2^{f (n) - n - 1}$ . Indeed, on the one hand, for all $m ⩾ n$ , $K (β_{s (m)} ↾ f (n)) < f (n) - n$ , and, if $m ⩾ g (f (n))$ , then, due to $f (n) > n$ and the assumption that g is increasing, $m ⩾ g (f (n)) > g (n) ⩾ n$ . On the other hand, it is clear that for any $k > 0$ there are at most $2^{k - 1}$ strings w with $K (w) < k$ .
Putting these observations together we conclude $\begin{array}{r} | 0 . u_{s (g (f (n)))} - 0 . α | ⩽ (| U_{n} | + 1) \cdot 2^{- f (n)} ⩽ (| V_{n} | + 1) \cdot 2^{- f (n)} ⩽ (2^{f (n) - n - 1} + 1) \cdot 2^{- f (n)} ⩽ 2^{- n} . \end{array}$ This ends the proof of the first assertion and, thus, the proof of the theorem. □

9. Promptly simple sets and nearly computable numbers

In this section we are going to strengthen the following result by Downey and LaForte.

Proposition 9.1 (Downey and LaForte [7, Theorem 11]).

If a real number is left-computable and nearly computable then it does not have promptly simple Turing degree.

A set $A \subseteq N$ is called simple if it is c.e., coinfinite, and its complement does not contain an infinite c.e. subset. This notion was strengthened as follows by Maass [15].

Definition 9.2.
A set $A \subseteq N$ is called promptly simple if it is c.e., coinfinite, and if there is a computable function $h : N \to N$ such that, for every $e \in N$ , if $W_{e}$ is infinite then there exist $s > 0$ and $x \in N$ such that $x \in (W_{e} [s] ∖ W_{e} [s - 1]) \cap A [h (s)]$ .

Clearly, every promptly simple set is simple. A Turing degree is called computably enumerable or c.e. if it contains a c.e. set. A Turing degree is called promptly simple if it contains a promptly simple set. Clearly, every promptly simple Turing degree is a c.e. Turing degree.

For a real number β, by the Turing degree of β we mean the Turing degree of the unique coinfinite set $B \subseteq N$ with $(β - 0 . B) \in Z$ . Note that this is identical with the Turing degree of the set ${n \in N : ν_{Q} (n) < β}$ . Hence, the Turing degree of any left-computable number is a c.e. Turing degree.

As within the c.e. Turing degrees the set of all promptly simple Turing degrees is upwards closed with respect to Turing reduction [22, Corollary XIII.1.9] (that means, if a is a promptly simple Turing degree and b is a c.e. Turing degree with $a ⩽_{T} b$ then b is a prompty simple Turing degree as well), Proposition 9.1 by Downey and LaForte [7, Theorem 11] is equivalent to the following assertion:
If β is a left-computable and nearly computable number then there does not exist a promptly simple set A with $A ⩽_{T} β$ .

We are going to show that this assertion can be strengthened as follows: One can omit the assumption that the real number is left-computable.
Theorem 9.3.
If β is a nearly computable number then there does not exist a promptly simple set A with $A ⩽_{T} β$ .

For the proof the following definition and the following observation due to Ambos-Spies et al. [1] will be useful.
Definition 9.4.
A sequence $B : N \to P (N)$ of finite subsets $B_{n} \subseteq N$ is called a strong array if the function $b : N \to N$ defined by $b (n) : = \sum_{i \in B_{n}} 2^{i}$ is computable.
Lemma 9.5 (Slowdown Lemma, Ambos-Spies et al. [1]).

Let ${(U_{k})}_{k}$ be a strong array such that, for all $e, n$ , $U_{⟨ e, n ⟩} \subseteq U_{⟨ e, n + 1 ⟩}$ . For each e, let $U_{e} : = ⋃_{n \in N} U_{⟨ e, n ⟩}$ . Then there exists a total computable function $g : N \to N$ such that, for all e, $W_{g (e)} = U_{e}$ and, for all $e \in N$ and $s > 0$ , $(U_{⟨ e, s ⟩} ∖ U_{⟨ e, s - 1 ⟩}) \cap W_{g (e)} [s] = \emptyset$ .

In the rest of this section we identify a subset $A \subseteq N$ with its characteristic function $χ_{A} : N \to {0, 1}$ . So, for a number $n \in N$ we write $\begin{array}{r} A (n) : = χ_{A} (n) = \{\begin{array}{ll} 0 & if n \notin A, \\ 1 & if n \in A, \end{array} \end{array}$ and $A ↾ n$ is the binary string $A (0) \dots A (n - 1)$ of length n. In the proof of Theorem 9.3 we shall use the following characterization of Turing reducibility.

Lemma 9.6.

For two sets $A, B \subseteq N$ the following two conditions are equivalent.

$A ⩽_{T} B$ .

There exists a c.e. set $C \subseteq {0, 1}^{*} \times N \times {0, 1}$ with the following properties.

For any triples $(w, n, z), (w^{'}, n^{'}, z^{'}) \in C$ , if w is a prefix of $w^{'}$ and $n = n^{'}$ then $z = z^{'}$ .

For any n there exists a number ℓ with $(B ↾ ℓ, n, A (n)) \in C$ .

Proof of Theorem 9.3.

Let us consider a computably approximable number β and a promptly simple set A with $A ⩽_{T} β$ . We can assume without loss of generality $0 ⩽ β < 1$ . Let $B \subseteq N$ be the uniquely determined set with $0 . B = β$ (note that A is simple, hence, undecidable; therefore, $A ⩽_{T} β$ implies that β is not computable; therefore there exists exactly one set $B \subseteq N$ with $β = 0 . B$ , and this set is undecidable, hence, coinfinite). It is sufficient to show that β is not nearly computable. Let us consider an arbitrary computable sequence ${(b_{n})}_{n}$ of rational numbers that converges to β. It is sufficient to show that this sequence is not nearly computably Cauchy. Hence, it is sufficient to show that there exists an increasing computable function $r : N \to N$ such that the sequence ${(b_{r (n + 1)} - b_{r (n)})}_{n}$ (which converges to 0, of course) does not converge computably. Thus, it is sufficient to construct a computable increasing function $r : N \to N$ such that for all $e \in N$ the function $φ_{e}$ is not a modulus of convergence of the sequence ${(b_{r (n + 1)} - b_{r (n)})}_{n}$ . So, we wish to construct a computable increasing function $r : N \to N$ such that for all $e \in N$ the following requirement $R_{e}$ is satisfied: $\begin{array}{r} R_{e} : if φ_{e} is total then (\exists m, n \in N) (m ⩾ φ_{e} (n) and | b_{r (m + 1)} - b_{r (m)} | > 2^{- n}) . \end{array}$

Let $h : N \to N$ be a computable function that shows the prompt simplicity of A, that is, a function such that for every $e \in N$ , if $W_{e}$ is infinite then there exist $s > 0$ and $x \in N$ such that $x \in (W_{e} [s] ∖ W_{e} [s - 1]) \cap A [h (s)]$ . Let $C \subseteq {0, 1}^{*} \times N \times {0, 1}$ be a c.e. set as in Lemma 9.6 applied to A and to the coinfinite set B with $0 . B = β$ , that is, C shows as in Lemma 9.6 that $A ⩽_{T} β$ .

We shall construct the desired function r in a sequence of stages; the value $r (s)$ will be defined in stage s. At the same time we are going to construct a certain strong array ${(U_{k})}_{k}$ such that for all $e, n$ , $U_{⟨ e, n ⟩} \subseteq U_{⟨ e, n + 1 ⟩}$ . We shall write $U_{e} [s]$ for $U_{⟨ e, s ⟩}$ . Let $g : N \to N$ be a function as in the Slowdown Lemma, applied to ${(U_{k})}_{k}$ .

We start in stage $s = 0$ by setting $r (0) : = 0$ and $U_{e} [0] : = \emptyset$ , for all $e \in N$ . Then we continue with stage $s = 1$ .

Let us now describe what we do in a stage $s > 0$ . First we check whether there exists a number $e < s$ with the following properties:

${0, \dots, | U_{e} [s - 1] |} \subseteq W_{e} [s]$ (note that the requirement $R_{e}$ is nontrivial only for numbers e with $W_{e} = N$ ),

$(\forall n \in W_{e} [s]) (\forall m < s - 1) (if m ⩾ φ_{e} (n) then | b_{r (m + 1)} - b_{r (m)} | ⩽ 2^{- n})$ (this means that the restriction of $φ_{e}$ to the set $W_{e} [s]$ looks as if $φ_{e}$ might be a modulus of convergence of the sequence ${(b_{r (m + 1)} - b_{r (m)})}_{m}$ ),

there exists a binary string w satisfying $| w | ⩽ s$ and $0 . w < b_{r (s - 1)} < 0 . w + 2^{- | w |}$ (this implies that $0 . w$ is a prefix of the binary representation of $b_{r (s - 1)}$ ) and such that the smallest number y with $2^{- y} < min {b_{r (s - 1)} - 0 . w, 0 . w + 2^{- | w |} - b_{r (s - 1)}}$ is an element of $W_{e} [s]$ and satisfies $φ_{e} (y) < s$ and such that there exists a number $x \in {0, 1, \dots, s} ∖ U_{e} [s - 1]$ with $A (x) [s] = 0$ and $(w, x, 0) \in C [s]$ .

If there does not exist such a number e then we set

r (s) : = r (s - 1) + 1

and

U_{\tilde{e}} [s] : = U_{\tilde{e}} [s - 1]

, for all

\tilde{e}

, we end this stage and continue with stage

s + 1

If there exists such a number e then we fix the smallest such e, say that $R_{e}$ receives attention at stage s and proceed as follows. First we fix the shortest binary string w satisfying the third condition above. Then we fix the smallest number x as in the third condition above. We enumerate x into the set $U_{e}$ , that is, we set $U_{e} [s] : = U_{e} [s - 1] \cup {x}$ (and we set $U_{\tilde{e}} [s] : = U_{\tilde{e}} [s - 1]$ for all $\tilde{e} \in N ∖ {e}$ ). We search for the smallest number t with $x \in W_{g (e)} [t]$ (note that the Slowdown Lemma, which ensures the existence of the function g, is proved by the Recursion Theorem; for a justification why we can use the function g here, in the definition of ${(U_{k})}_{k}$ , see [22, Chapter II.3]). Such a number t exists, and it is strictly greater than s. Now we check whether $x \in A [h (t)]$ . If this is not the case then we say that the current attempt to satisfy the requirement $R_{e}$ has failed, we set $r (s) : = r (s - 1) + 1$ , and we continue with stage $s + 1$ .

If, on the other hand, $x \in A [h (t)]$ then let u be the smallest number strictly greater than $r (s - 1)$ such that there exists a binary string $\tilde{w}$ with $0 . \tilde{w} < b_{u} < 0 . \tilde{w} + 2^{- | \tilde{w} |}$ and $(\tilde{w}, x, 1) \in C [u]$ . We set $r (s) : = u$ and continue with stage $s + 1$ . In this case we say that the current attempt to satisfy the requirement $R_{e}$ was successful. This ends the description of the algorithm.

We come to the verification. It is clear that the constructed function $r : N \to N$ is increasing and computable. We have to show that it satisfies all requirements $R_{e}$ . For the sake of a contradiction, let us assume that this is not the case. Let e be the smallest number such that the requirement $R_{e}$ is not satisfied. Then $φ_{e}$ is a total function and a modulus of convergence of the sequence ${(b_{r (m + 1)} - b_{r (m)})}_{m}$ .

First, we show that the requirements $R_{d}$ with $d < e$ will receive attention only finitely often. For d such that $φ_{d}$ is not a total function this is due to the condition ${0, \dots, | U_{d} [s - 1] |} \subseteq W_{d} [s]$ (if $R_{d}$ would receive attention infinitely often then $U_{d}$ would be infinite and this condition would imply $W_{d} = N$ ). And if $d < e$ has the property that $φ_{d}$ is total and $R_{d}$ is satisfied then there exist $m, n$ with $m ⩾ φ_{d} (n)$ and $| b_{r (m + 1)} - b_{r (m)} | > 2^{- n}$ , and for sufficiently large s we have $n \in W_{d} [s]$ and $m < s - 1$ . So, we have shown that the requirements $R_{d}$ with $d < e$ receive attention only finitely often.

Next, we show that $R_{e}$ will receive attention infinitely often. For the sake of a contradiction let us assume that this is not the case and that there is some $s_{0} > 0$ such that no $R_{d}$ with $d ⩽ e$ will receive attention at any stage $s ⩾ s_{0}$ . Then for $s ⩾ s_{0}$ the set $U_{e} [s - 1]$ does not change anymore, hence, $U_{e} [s - 1] = U_{e}$ , hence, for sufficiently large s we have ${0, \dots, | U_{e} [s - 1] |} \subseteq W_{e} [s]$ . As we assume that $R_{e}$ is not satisfied, the function $φ_{e}$ is a total function and a modulus of convergence of the sequence ${(b_{r (m + 1)} - b_{r (m)})}_{m}$ . Hence, the second condition above is satisfied as well. As A is coinfinite let us consider some $x \in N ∖ A$ that is not an element of $U_{e}$ . Then $A (x) = 0$ , hence, for all s, $A (x) [s] = 0$ , and for sufficiently large s we have $x ⩽ s$ , hence, $x \in {0, 1, \dots, s} ∖ U_{e} [s - 1]$ . There exists a number ℓ such that $(B ↾ ℓ, x, 0) \in C$ . We set $w : = B ↾ ℓ$ . For sufficiently large s we have $| w | ⩽ s$ and $(w, x, 0) \in C [s]$ . As the sequence ${(b_{k})}_{k}$ converges to $β = 0 . B$ for k tending to ∞ and B is not decidable, for sufficiently large s we have $0 . w < b_{r (s - 1)} < 0 . w + 2^{- | w |}$ . Furthermore, for sufficiently large s the smallest number y with $2^{- y} < min {b_{r (s - 1)} - 0 . w, 0 . w + 2^{- | w |} - b_{r (s - 1)}}$ is an element of $W_{e} [s]$ (because $W_{e} = N$ ) and satisfies $φ_{e} (y) < s$ . We conclude that $R_{e}$ will receive attention at a sufficiently large stage $s ⩾ s_{0}$ . Contradiction. We have shown that $R_{e}$ will receive attention infinitely often.

This implies that the set $U_{e} = W_{g (e)}$ is infinite. Now the fact that A is promptly simple and that the function h shows this imply that there exist numbers x and $t > 0$ with $x \in (W_{g (e)} [t] ∖ W_{g (e)} [t - 1]) \cap A [h (t)]$ . Then there must exist some s with $0 < s < t$ such that $x \in U_{e} [s] ∖ U_{e} [s - 1]$ , and this is possible only if $R_{e}$ receives attention in stage s. Hence, there must exist a binary string w with $| w | ⩽ s$ and $0 . w < b_{r (s - 1)} < 0 . w + 2^{- | w |}$ and such that the smallest number y with $2^{- y} < min {b_{r (s - 1)} - 0 . w, 0 . w + 2^{- | w |} - b_{r (s - 1)}}$ is an element of $W_{e} [s]$ and satisfies $φ_{e} (y) < s$ , and such that x is the smallest number in ${0, 1, \dots, s} ∖ U_{e} [s - 1]$ satisfying $A (x) [s] = 0$ and $(w, x, 0) \in C [s]$ . On the other hand, as $x \in A [h (t)]$ , in the algorithm we define $r (s)$ is such a way that for some binary string $\tilde{w}$ we have $0 . \tilde{w} < b_{r (s)} < 0 . \tilde{w} + 2^{- | \tilde{w} |}$ and $(\tilde{w}, x, 1) \in C [r (s)]$ . The two facts $(w, x, 0) \in C$ and $(\tilde{w}, x, 1) \in C$ imply that none of the two strings w and $\tilde{w}$ can be a prefix of the other one. Hence, the intervals $(0 . w, 0 . w + 2^{- | w |})$ and $(0 . \tilde{w}, 0 . \tilde{w} + 2^{- | \tilde{w} |})$ have empty intersection. As the distance of $b_{r (s - 1)}$ , which is an element of the interval $(0 . w, 0 . w + 2^{- | w |})$ , to the endpoints of this interval is larger than $2^{- y}$ and as $b_{r (s)}$ is an element of the interval $(0 . \tilde{w}, 0 . \tilde{w} + 2^{- | \tilde{w} |})$ , which is disjoint from the interval $(0 . w, 0 . w + 2^{- | w |})$ , we conclude that $| b_{r (s - 1)} - b_{r (s)} | > 2^{- y}$ . But in combination with $φ_{e} (y) < s$ this means that $R_{e}$ is satisfied after this stage s. Contradiction. □

Corollary 6.9 leaves open the question which real numbers can appear as values $f (α)$ for nearly computable numbers α and computable real functions f. Here Theorem 9.3 restricts the possibilities.

Corollary 9.7.

If β is a nearly computable number and $f : \subseteq R \to R$ a computable function with $β \in dom (f)$ then there does not exist a promptly simple set A with $A ⩽_{T} f (β)$ .

Note that there are promptly simple sets [22, Exercise XIII.1.10].

Proof of Corollary 9.7.

If $f : \subseteq R \to R$ is a computable function and $β \in dom (f)$ then $f (β) ⩽_{T} β$ (compare Ko [13]). Hence, if there were a promptly simple set A with $A ⩽_{T} f (β)$ then the transitivity of $⩽_{T}$ would give us $A ⩽_{T} β$ , and by Theorem 9.3 this is not possible if β is nearly computable. □

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