There exists a non-recursively enumerable set $\{n \in \mathbb{N}: \varphi(n)\}$ which is co-recursively enumerable, where the formula $\varphi(n)$ is short and can be easily translated into a first-order formula in Peano arithmetic

TL;DR

This paper demonstrates the existence of a non-recursively enumerable set that is co-recursively enumerable, defined by a short formula translatable into Peano arithmetic, with implications for computability and Diophantine equations.

Contribution

It constructs a specific non-recursively enumerable yet co-recursively enumerable set using short formulas and G"odel's $eta$ function, advancing understanding of computability in number theory.

Findings

The set $ exists n$ satisfying formula (1) is co-recursively enumerable.

Functions $f(n)$ and $eta(n)$ are computable in the limit and dominate all computable functions.

A short program converges to the functions $f(n)$ and $eta(n)$, illustrating their computability in the limit.

Abstract

We prove that the sets $\{n \in N: n$ satisfies formula (1)$\}$ and $\{n \in N: n$ does not satisfy formula (2)$\}$ are not recursively enumerable. We prove that these sets are co-recursively enumerable. $(1)~\exists p,q \in N~((2n=(p+q)(p+q+1)+2q) \wedge \forall (x_0,...,x_p) \in N^{p+1}~\exists (y_0,...,y_p) \in\{0,...,q\}^{p+1}~\forall i,j,k \in \{0,...,p\}~(((x_j+1=x_k) \Rightarrow (y_j+1=y_k)) \wedge ((x_i \cdot x_j=x_k) \Rightarrow (y_i \cdot y_j=y_k))))$. $(2)~\exists (y_0,...,y_n) \in N^{n+1}~\forall i,j,k \in \{0,...,n\}~(((2^{\textstyle 2^{\textstyle 2^j \cdot 3^k}}+1$ divides $n) \Rightarrow (y_j+1=y_k)) \wedge ((2^{\textstyle 2^{\textstyle 2^i \cdot 3^j \cdot 5^{k+1}}}+1$ divides $n) \Rightarrow (y_i \cdot y_j=y_k)))$. By using G\"odel's $\beta$ function, we can easily translate formula (1) into a first-order formula in Peano arithmetic. For $n \in N$, let $E_n=\{1=x_k,~x_i+x_j=x_k,~x_i \cdot x_j=x_k:~i,j,k \in \{0,...,n\}\}$. 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 $\{0,...,b\}^{n+1}$. The function $f:N \to N$ is computable in the limit and eventually dominates every computable function $g:N \to N$. We present a short program which for $n \in N$ prints the sequence $\{f_i(n)\}_{i=0}^\infty$ of non-negative integers converging to $f(n)$. For $n \in N$, $\gamma(n)$ denotes the smallest $b \in N$ such that if a system of equations $S \subseteq E_n$ has a unique solution in $N^{n+1}$, then this solution belongs to $\{0,...,b\}^{n+1}$. The function $\gamma:N \to N$ is computable in the limit and eventually dominates every function $\delta:N \to N$ with a single-fold Diophantine representation. We present a short program which for $n \in N$ prints the sequence $\{\gamma_i(n)\}_{i=0}^\infty$ of non-negative integers converging to $\gamma(n)$.