I want to coin a notion for reducibility of theories. Generally this goes like that: if we have two equi-interpretable theories $T;Q$ and it is harder to interpret $T$ in $Q$ than to interpret $Q$ in $T$; then we are to say that: theory $T$ is reducible to theory $Q$.
For example lets take theory $ZF^{fin}$ to mean the theory $ZF -\text{Infinity} + \text{ all sets are finite}$. Now if we have: $$ \forall F [(F:PA \approx ZF^{fin} ) \to \exists G ((G: ZF^{fin} \approx PA) \land G \leq F)]$$; where $I: T \approx Q$ means, $I$ is an interpretation of $Q$ in $T$ , and $\leq$ means easier than or equal in hardness. Then since $PA$ and $ZF^{fin}$ are equi-interpretable theories, then we are to say that: $$ZF^{fin} \text { is reducible to } PA$$ If the opposite is not true, then we have $$\neg (PA \text { is reducible to } ZF^{fin})$$, then we'd say in this case that: $$ ZF^{fin} \text { is strictly reducible to } PA$$
Is there a mathematical criterion for hardness of interpret-ability of theories? For example which is considered harder? to interpret $ZF + \neg C$ in $ZFC$ or the opposite?
Can we even in some remote sense consider reducibility in this particular sense to be a crierion of truth? for example if $T$ is strictly reducible to $Q$, then $Q$ is the true theory?
