TITLE: Is it possible that two theories be equiconsistent, with Peano Arithmetic not able to prove this?
QUESTION [6 upvotes]: Do there exist first-order theories that are are equiconsistent, but which cannot be proven to be equiconsistent using Peano Arithmetic? (I hope not.)

REPLY [4 votes]: There is a phenomenon that happens when we look at questions like these, which often makes the answers uninteresting. 
Let $S = \{\phi_n\}$ be the usual effective enumeration of the axioms of PA. Define a theory $T = \{\psi_1, \psi_2, \ldots\}$ where $\psi_i = \phi_i$ unless $i$ is a coded proof of $0=1$ from the axioms of ZFC, in which case $\psi_i$ is $0=1$.  Note that $T$ is an effective theory for which we can write down a single, fixed program to enumerate the axioms in the manner just described. Thus, when we formalize $T$ inide PA or ZFC, we do so by using this particular program to enumerate axioms of $T$ to make the Con predicate for $T$.
Now ZFC is consistent, so in fact $T$ is the same theory as $S$, and hence they are equiconsistent. But ZFC cannot prove $S$ is the same theory as $T$, because ZFC cannot prove that ZFC is consistent, and the definition of $T$ leverages this. Thus ZFC can prove $\operatorname{Con}(S)$ but not $\operatorname{Con}(T)$, so ZFC cannot prove that $S$ and $T$ are equiconsistent.
By tweaking the original theories $S$ and $T$, we can produce examples where ZFC proves the equiconsistency, but PA does not, or where PA proves the equiconsistency, but a weaker effective theory of arithmetic $W$ does not (where the consistency of $W$ is provable in PA). 
This phenomenon arises in many places when we begin to study formalized theories within ZFC or PA. Any time we quantify over all effective theories, this includes theories such as $T$ which are well behaved in the standard model but do strange things when formalized and then viewed in a nonstandard model. For this reason, it is usually more interesting to look at equiconsistency for natural theories with their canonical axiomatizations rather than with arbitrary axiomatizations.