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



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


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....) 



Is $PRA$ + $TI({\epsilon_0})$ mutually interpretable with some theory in the language of set theory?
Yes, the consistency of "ZFC with the axiom of infinity replaced by its negation" is provable in "PRA + TI($\epsilon\_0$)". Technically one has to also show that "PRA + TI($\epsilon\_0$)" can prove the equiconsistency result (since it already proves the consistency of PA), but these are fairly natural theories so that shouldn't be a problem.


I'm not aware of a natural fragment of ZFC equiconsistent with "PRA + TI($\epsilon\_0$)", and can't think of any reason there would be one. It's sort of remarkable that ZFC has a fragment which matches up perfectly with PA (which is mostly a reflection of how PA is a very natural theory).


