Finding the limit of incompleteness I

In this paper, we examine the limit of applicability of G\"{o}del's first incompleteness theorem ($\sf G1$ for short). We first define the notion "$\sf G1$ holds for the theory $T$". This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\sf G1$ holds. To approach this question, we first examine the following question: is there a theory $T$ such that Robinson's $\mathbf{R}$ interprets $T$ but $T$ does not interpret $\mathbf{R}$ (i.e. $T$ is weaker than $\mathbf{R}$ w.r.t. interpretation) and $\sf G1$ holds for $T$? In this paper, we show that there are many such theories based on Je\v{r}\'{a}bek's work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle$, we can construct a r.e. theory $U_{\langle A,B\rangle}$ such that $U_{\langle A,B\rangle}$ is weaker than $\mathbf{R}$ w.r.t. interpretation and $\sf G1$ holds for $U_{\langle A,B\rangle}$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf{0}< \mathbf{d}<\mathbf{0}^{\prime}$, there is a theory $T$ with Turing degree $\mathbf{d}$ such that $\sf G1$ holds for $T$ and $T$ is weaker than $\mathbf{R}$ w.r.t. Turing reducibility. As a corollary, based on Shoenfield's work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which $\sf G1$ holds.