  This work is motivated from finding the limit of the applicability of the
first incompleteness theorem ($\sf G1$). A natural question is: can we find a
minimal theory for which $\sf G1$ holds? The answer of this question depends on
our definition of minimality. We examine the Turing degree structure of
recursively enumerable (RE) theories for which $\sf G1$ holds and the
interpretation degree structure of RE theories weaker than the theory
$\mathbf{R}$ with respect to interpretation for which $\sf G1$ holds. We answer
all questions about these two structures in our past work and prove more
results about them. It is known that there are no minimal essentially
undecidable theories with respect to interpretation. We generalize this result
and give two general characterizations which tell us under what conditions
there are no minimal RE theories having some property with respect to
interpretation. Finally, as an application, we propose the theory version of
recursively inseparable theories, ${\sf tRI}$ theories, and show that there are
no minimal ${\sf tRI}$ theories with respect to interpretation: for any ${\sf
tRI}$ theory, we can effectively find a ${\sf tRI}$ theory which is strictly
weaker with respect to interpretation.
