As is well known, the following theory is equiconsistent with $PA$:

$ZFC$ with the axiom of infinity replaced by its negation.

Since this theory is equiconsistent with $PA$, it would seem reasonable to infer (wouldn't it?) that the consistency of '$ZFC$ with the axiom of infinity replaced by its negation' could be provable in "$PRA$ + $TI({\epsilon_0})$.

So what 'theory of sets'(?) is mutually interpretable with "$PRA$ + $TI({\epsilon_0})$? Also, can one define a notion of forcing in the aforementioned theory?

(If this seems too silly a question, please let me know and I will delete....)

nota fragment of ZFC (it is even inconsistent with ZFC). A fragment of ZFC equiconsistent with PA is ZFC with the axiom of infinitydropped. $\endgroup$ – Emil Jeřábek Dec 20 '16 at 15:44