The other principles

If the theory under consideration is $\mathit{PA}$ with full induction, more principles hold. We consider Montagna's principle $M : \ \ \alpha \rhd \beta \rightarrow \alpha \wedge \sigma \rhd \beta \wedge \sigma$ for every $\sigma \in \Sigma^0_1$. This can be seen to hold for $\mathit{PA}$using the following fact. Over $\mathit{PA}$ one knows that $\alpha \rhd \beta$ iff every $\mathit{PA}$-model of $\alpha$ has an end extension which is a $\mathit{PA}$-model of $\beta$. So if the $\Sigma^0_1$-sentence $\sigma$holds in a $\mathit{PA}$-model M of $\alpha$, there must be a witness in this model for $\sigma$. This witness serves as a witness as well in any end extension of M. We say that $\sigma$ is being preserved under taking end extensions. So there exists an end extension for $\beta \wedge \sigma$, hence $\alpha \wedge \sigma \rhd \beta \wedge \sigma$.

If T, the theory under consideration, is finitely axiomatizable, the situation gets a lot easier. The sentence $\forall y({\mathit Ax}_T(y)\rightarrow \Box (\alpha \rightarrow y^J))$ can just be replaced by $\Box (\alpha \rightarrow \tau^J)$, where $\tau$ is the conjunction of all the axioms of T. Consequently the sentence (+), expressing formal interpretability, becomes a $\Sigma^0_1$-sentence and hence can be ``boxed up'', using so-called provable $\Sigma^0_1$-completeness already used by Gödel. By doing so one obtains the so-called persistence principle $P : \ \ \alpha \rhd \beta \rightarrow \Box (\alpha \rhd \beta)$. As $I\Sigma_n$ is finitely axiomatizable for all $n\in \omega$, we have that P is an interpretability principle for $I\Sigma_n$ for all $n\in \omega$.