A Non-Recursively Enumerable Subset of <em>N </em>Which Has {a Short} Description in Terms of Arithmetic

Apoloniusz Tyszka

doi:10.20944/preprints202508.0363.v10

Submitted:

25 May 2026

Posted:

27 May 2026

You are already at the latest version

Abstract

Let F(x, n) denote the formula

\exists ab \forall i \leq n \exists s wpq \forall j ν \exists e g {(s + w)^{2} + 3 w + s = 2 i \land < [j = w \land ν = q] ν [j = 3 i \land ν = p + q] \lor [j = s \land (ν = p \lor (i = n \land ν = q + x))] ν [j = 3 i + 1 \land

ν = pq] \to a = ν + e + e jb \land ν + g = jb >}

from J. P. Jones’ article published in 1978. From the results of this article, it follows that the set {n ∈ N : ¬F(n, n)} is non-recursively enumerable and co-recursively enumerable. We prove that the set W = {n ∈ N : ∃p, q ∈ N ((2n = (p + q)(p + q + 1) + 2q) ∧ ∀(x0, . . . , xp) ∈ Np+1 ∃(y0, . . . , yp) ∈ {0, . . . , q}p+1 ((∀j, k ∈ {0, . . . , p} (xj + 1 = xk ⇒ yj + 1 = yk)) ∧ (∀i, j, k ∈ {0, . . . , p} (xi · xj = xk ⇒ yi · yj = yk))))} is not recursively enumerable. We prove that the set N \W is recursively enumerable. Let β : N³ → N denote Gödel’s β function. For x₁, x₂, x₃∈ N, β(x₁, x₂, x₃) equals the remainder after integer division of x₁by 1 + (x₃+ 1) · x₂. We prove that the set W consists of all n ∈ N such that ∀u, v ∈ N ∃a, b, p, q ∈ N ((2n = (p + q)(p + q + 1) + 2q) ∧ ∀i, j, k ∈ {0, . . . , p} ((β(a, b, i) ⩽ q) ∧ (β(u, v, j) + 1 = β(u, v, k) ⇒ β(a, b, j) + 1 = β(a, b, k)) ∧ (β(u, v, i) · β(u, v, j) = β(u, v, k) ⇒ β(a, b, i) · β(a, b, j) = β(a, b, k)))) The above formula can be easily translated into a formula in Peano arithmetic.

Keywords:

co-recursively enumerable set

;

eventual domination

;

Gödel’s β function

;

limit-computable function

;

recursively enumerable set

Subject:

Computer Science and Mathematics - Logic

MSC: 03D25

Let

F (x, n)

denote the formula

from [2] (p. 336). From the results of [2], it follows that the set

{n \in N : \neg F (n, n)}

is non-recursively enumerable and co-recursively enumerable. In this article, we define another non-recursively enumerable subsets of

N

which have a short description in terms of arithmetic. Semi-algorithms differ from algorithms, as they may not terminate.

Definition 1

(cf. [4], pp. 233–235). A computation in the limit of a function

f : N \to N

is a semi-algorithm which takes as input a non-negative integer n and for every

m \in N

prints a non-negative integer

ξ (n, m)

such that

lim_{m \to \infty} ξ (n, m) = f (n)

.

By Definition 1, a function

f : N \to N

is computable in the limit when there exists an infinite computation which takes as input a non-negative integer n and prints a non-negative integer on each iteration and prints

f (n)

on each sufficiently high iteration.

For

n \in N

, let

E_{n} = {1 = x_{k}, x_{i} + x_{j} = x_{k}, x_{i} \cdot x_{j} = x_{k} : i, j, k \in {0, \dots, n}}

For

n \in N

,

f (n)

denotes the smallest

b \in N

such that if a system of equations

S \subseteq E_{n}

has a solution in

N^{n + 1}

, then S has a solution in

{0, \dots, b}^{n + 1}

. The function

f : N \to N

is computable in the limit and eventually dominates every computable function

g : N \to N

, see [6]. The term "dominated" in the title of [6] means "eventually dominated".

Theorem 1

([6]). Figure 1 shows a semi-algorithm which computes

f (n)

in the limit.

Definition 2.

An approximation of a tuple

(x_{0}, \dots, x_{n}) \in N^{n + 1}

is a tuple

(y_{0}, \dots, y_{n}) \in N^{n + 1}

such that

(\forall k \in {0, \dots, n} (1 = x_{k} \Rightarrow 1 = y_{k})) \land

(\forall i, j, k \in {0, \dots, n} (x_{i} + x_{j} = x_{k} \Rightarrow y_{i} + y_{j} = y_{k})) \land

(\forall i, j, k \in {0, \dots, n} (x_{i} \cdot x_{j} = x_{k} \Rightarrow y_{i} \cdot y_{j} = y_{k}))

Figure 2 shows a simpler semi-algorithm which computes

f (n)

in the limit.

Lemma 1.

For every

n, m \in N

, the number printed by Figure 2 does not exceed the number printed by Figure 1.

Proof.

For every

(a_{0}, \dots, a_{n}) \in {0, \dots, m}^{n + 1}

,

E_{n} \supseteq {1 = x_{k} : (k \in {0, \dots, n}) \land (1 = a_{k})} \cup

{x_{i} + x_{j} = x_{k} : (i, j, k \in {0, \dots, n}) \land (a_{i} + a_{j} = a_{k})} \cup

{x_{i} \cdot x_{j} = x_{k} : (i, j, k \in {0, \dots, n}) \land (a_{i} \cdot a_{j} = a_{k})}

□

Lemma 2.

For every

n, m \in N

, the number printed by Figure 1 does not exceed the number printed by Figure 2.

Proof.

Let

n, m \in N

. For every system of equations

S \subseteq E_{n}

, if

(a_{0}, \dots, a_{n}) \in {0, \dots, m}^{n + 1}

and

(a_{0}, \dots, a_{n})

solves S, then

(a_{0}, \dots, a_{n})

solves the following system of equations:

{1 = x_{k} : (k \in {0, \dots, n}) \land (1 = a_{k})} \cup

{x_{i} + x_{j} = x_{k} : (i, j, k \in {0, \dots, n}) \land (a_{i} + a_{j} = a_{k})} \cup

{x_{i} \cdot x_{j} = x_{k} : (i, j, k \in {0, \dots, n}) \land (a_{i} \cdot a_{j} = a_{k})}

□

Theorem 2.

For every

n, m \in N

, Figure 1 and Figure 2 print the same number.

Proof.

It follows from Lemmas 1 and 2. □

Corollary 1.

For every

n, m \in N

, Figure 1 and Figure 2 print the smallest

b \in {0, \dots, m}

such that every tuple

(x_{0}, \dots, x_{n}) \in {0, \dots, m}^{n + 1}

possesses an approximation in

{0, \dots, b}^{n + 1}

.

Theorem 3.

For every

n \in N

,

f (n)

is the smallest

b \in N

such that every tuple

(x_{0}, \dots, x_{n}) \in N^{n + 1}

possesses an approximation in

{0, \dots, b}^{n + 1}

.

Proof.

It follows from Theorem 1 and Corollary 1. □

Theorem 4.

No algorithm takes as input non-negative integers n and m and decides whether or not

\forall (x_{0}, \dots, x_{n}) \in N^{n + 1} \exists (y_{0}, \dots, y_{n}) \in {0, \dots, m}^{n + 1}

((\forall k \in {0, \dots, n} (1 = x_{k} \Rightarrow 1 = y_{k})) \land

(\forall i, j, k \in {0, \dots, n} (x_{i} + x_{j} = x_{k} \Rightarrow y_{i} + y_{j} = y_{k})) \land

(\forall i, j, k \in {0, \dots, n} (x_{i} \cdot x_{j} = x_{k} \Rightarrow y_{i} \cdot y_{j} = y_{k})))

Proof.

Since the function f is not computable, it follows from Theorem 3. □

Lemma 3

([3]). The function

N^{2} ∋ (p, q) \to \frac{1}{2} (p + q) (p + q + 1) + q \in N

is bijective.

Theorem 5.

No algorithm takes as input a non-negative integer n and decides whether or not

\exists p, q \in N ((2 n = (p + q) (p + q + 1) + 2 q)) \land

\forall (x_{0}, \dots, x_{p}) \in N^{p + 1} \exists (y_{0}, \dots, y_{p}) \in {0, \dots, q}^{p + 1}

((\forall k \in {0, \dots, p} (1 = x_{k} \Rightarrow 1 = y_{k})) \land

(\forall i, j, k \in {0, \dots, p} (x_{i} + x_{j} = x_{k} \Rightarrow y_{i} + y_{j} = y_{k})) \land

(\forall i, j, k \in {0, \dots, p} (x_{i} \cdot x_{j} = x_{k} \Rightarrow y_{i} \cdot y_{j} = y_{k}))))

Proof.

It follows from Theorem 4 and Lemma 3. □

Let

T = {n \in N : \exists p, q \in N ((2 n = (p + q) (p + q + 1) + 2 q) \land

\forall (x_{0}, \dots, x_{p}) \in N^{p + 1} \exists (y_{0}, \dots, y_{p}) \in {0, \dots, q}^{p + 1}

((\forall k \in {0, \dots, p} (1 = x_{k} \Rightarrow 1 = y_{k})) \land

(\forall i, j, k \in {0, \dots, p} (x_{i} + x_{j} = x_{k} \Rightarrow y_{i} + y_{j} = y_{k})) \land

(\forall i, j, k \in {0, \dots, p} (x_{i} \cdot x_{j} = x_{k} \Rightarrow y_{i} \cdot y_{j} = y_{k}))))}

Theorem 6.

The set

N ∖ T

is recursively enumerable.

Proof.

For

i \in N

, let

p_{i}

denote the i-th prime number. Figure 3 shows a semi-algorithm which takes as input

n \in N

and terminates if and only if

n \in N ∖ T

. □

Theorem 7.

The set T is not recursively enumerable.

Proof.

It follows from Theorems 5 and 6. □

Lemma 4.

([5], p. 110). For non-negative integers, the equation

x + y = z

is equivalent to a system which consists of equations of the forms

v + 1 = w

and

u \cdot v = w

.

For

n \in N

,

h (n)

denotes the smallest

b \in N

such that if a system of equations

S \subseteq

{x_{j} + 1 = x_{k}, x_{i} \cdot x_{j} = x_{k} : i, j, k \in {0, \dots, n}}

has a solution in

N^{n + 1}

, then S has a solution in

{0, \dots, b}^{n + 1}

. From Lemma 4 and [6], it follows that the function

h : N \to N

is computable in the limit and eventually dominates every computable function

g : N \to N

.

Theorem 8.

No algorithm takes as input non-negative integers n and m and decides whether or not

\forall (x_{0}, \dots, x_{n}) \in N^{n + 1} \exists (y_{0}, \dots, y_{n}) \in {0, \dots, m}^{n + 1}

((\forall j, k \in {0, \dots, n} (x_{j} + 1 = x_{k} \Rightarrow y_{j} + 1 = y_{k})) \land

(\forall i, j, k \in {0, \dots, n} (x_{i} \cdot x_{j} = x_{k} \Rightarrow y_{i} \cdot y_{j} = y_{k})))

Proof.

It holds because the function h is not computable, and for every

n \in N

,

h (n)

is the smallest

b \in N

such that

\forall (x_{0}, \dots, x_{n}) \in N^{n + 1} \exists (y_{0}, \dots, y_{n}) \in {0, \dots, b}^{n + 1}

((\forall j, k \in {0, \dots, n} (x_{j} + 1 = x_{k} \Rightarrow y_{j} + 1 = y_{k})) \land

(\forall i, j, k \in {0, \dots, n} (x_{i} \cdot x_{j} = x_{k} \Rightarrow y_{i} \cdot y_{j} = y_{k})))

□

Theorem 9.

No algorithm takes as input a non-negative integer n and decides whether or not

\exists p, q \in N ((2 n = (p + q) (p + q + 1) + 2 q) \land

\forall (x_{0}, \dots, x_{p}) \in N^{p + 1} \exists (y_{0}, \dots, y_{p}) \in {0, \dots, q}^{p + 1}

((\forall j, k \in {0, \dots, p} (x_{j} + 1 = x_{k} \Rightarrow y_{j} + 1 = y_{k})) \land

(\forall i, j, k \in {0, \dots, p} (x_{i} \cdot x_{j} = x_{k} \Rightarrow y_{i} \cdot y_{j} = y_{k}))))

Proof.

It follows from Theorem 8 and Lemma 3. □

Let W denote the algorithmically undecidable subset of

N

considered in Theorem 9. Similarly as in Theorem 6, the set

N ∖ W

is recursively enumerable. Similarly as in Theorem 7, the set W is not recursively enumerable. Let

β : N^{3} \to N

denote Gödel’s

β

function, see [1]. For

x_{1}, x_{2}, x_{3} \in N

,

β (x_{1}, x_{2}, x_{3})

equals the remainder after integer division of

x_{1}

by

1 + (x_{3} + 1) \cdot x_{2}

.

Lemma 5

([1]). If

(d_{0}, \dots, d_{p}) \in N^{p + 1}

, then

\exists b, c \in N \forall l \in {0, \dots, p} β (b, c, l) = d_{l}

.

Theorem 10.

The formula that defines the set W can be easily translated into a formula in Peano arithmetic.

Proof.

By Lemma 5, the set W consists of all

n \in N

such that

\forall u, v \in N \exists a, b, p, q \in N ((2 n = (p + q) (p + q + 1) + 2 q) \land \forall i, j, k \in {0, \dots, p}

((β (a, b, i) ⩽ q) \land (β (u, v, j) + 1 = β (u, v, k) \Rightarrow β (a, b, j) + 1 = β (a, b, k)) \land

(β (u, v, i) \cdot β (u, v, j) = β (u, v, k) \Rightarrow β (a, b, i) \cdot β (a, b, j) = β (a, b, k))))

The above formula can be easily translated into a formula in Peano arithmetic. □

References

Gödel’s β function, https://en.wikipedia.org/wiki/G%C3%B6del%27s_%CE%B2_function.
J. P. Jones, Three universal representations of recursively enumerable sets, J. Symbolic Logic 43 (1978), no. 2, 335–351. [CrossRef]
Pairing function, https://en.wikipedia.org/wiki/Pairing_function.
R. I. Soare, Interactive computing and relativized computability, in: B. J. Copeland, C. J. Posy, and O. Shagrir (eds.), Computability: Turing, Gödel, Church and beyond, MIT Press, Cambridge, MA, 2013, 203–260.
A. Tyszka, A hypothetical upper bound on the heights of the solutions of a Diophantine equation with a finite number of solutions, Open Comput. Sci. 8 (2018), no. 1, 109–114. [CrossRef]
A. Tyszka, All functions $g : N \to N$ which have a single-fold Diophantine representation are dominated by a limit-computable function $f : N ∖ {0} \to N$ which is implemented inMuPADand whose computability is an open problem, in: Computation, cryptography, and network security (eds. N. J. Daras, M. Th. Rassias), Springer, Cham, 2015, 577–590. [CrossRef]

Figure 1. A semi-algorithm which computes

f (n)

in the limit.

Figure 1. A semi-algorithm which computes

f (n)

in the limit.

Figure 2. A simpler semi-algorithm which computes

f (n)

in the limit.

Figure 2. A simpler semi-algorithm which computes

f (n)

in the limit.

Figure 3. A semi-algorithm which takes as input

n \in N

and terminates if and only if

n \in N ∖ T

.

Figure 3. A semi-algorithm which takes as input

n \in N

and terminates if and only if

n \in N ∖ T

.

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A Non-Recursively Enumerable Subset of N Which Has {a Short} Description in Terms of Arithmetic

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