  We consider the foundational relation between arithmetic and set theory. Our
goal is to criticize the construction of standard arithmetic models as
providing grounds for arithmetic truth (even in a relative sense). Our method
is to emphasize the incomplete picture of both theories and treat models as
their syntactical counterparts. Insisting on the incomplete picture will allow
us to argue in favor of the revisability of the standard model interpretation.
We then show that it is hopeless to expect that the relative grounding provided
by a standard interpretation can resist being revisable. We start briefly
characterizing the expansion of arithmetic truth provided by the interpretation
in a set theory. Further, we show that, for every well-founded interpretation
of recursive extensions of PA in extensions of ZF, the interpreted version of
arithmetic has more theorems than the original. This theorem expansion is not
complete however. We continue by defining the coordination problem. The problem
can be summarized as follows. We consider two independent communities of
mathematicians responsible for deciding over new axioms for ZF and PA. How
likely are they to be coordinated regarding PAs interpretation in ZF? We prove
that it is possible to have extensions of PA not interpretable in a given set
theory ST. We further show that the probability of a random extension of
arithmetic being interpretable in ST is zero.
