Let's speak of the theory $\sf ZC + rank$ as the first order set theory with axioms of Extensionality, Separation, infinity, and choice (written as usual), plus iterative powers and foundation, those are:


**Iterative Power:** $\forall \text { ordinal } \alpha \exists x : x=P^\alpha(\emptyset)$


where $P^\alpha(\emptyset) = \bigcup \{P(P^\beta(\emptyset)) : \beta < \alpha\} $


**Foundation:** $\forall x \exists \text { ordinal } \alpha: x \in P^\alpha (\emptyset)$


Now ordinal is meant to be a Von Neumann ordinal.



> 
> is every extension of $\sf ZFC$ *interpretable* in a finite extension of $\sf ZC + rank$?
> 
> 
> is every finite extension of ZFC interpretable in a finite extension of ZC + rank?
> 
> 
> 



Is every extension of ZFC interpretable in a finite extension of ZC + rank?
$\let\res\restriction\def\N{\mathbb N}$By the discussion in comments, it is missing from the question that all the theories are assumed **consistent** (otherwise the answer is trivially yes, as everything is interpretable in the inconsistent extension) and **recursively axiomatizable** (otherwise the answer is trivially no, as there are $2^\omega$ pairwise incompatible complete extensions of ZFC, while only countably many of these are interpretable in any given consistent extension of ZC + rank).


With these caveats, the answer to both questions is **yes**, for general reasons that work for most other pairs of natural theories. In what follows, if $T$ is a theory with a fixed r.e. set of axioms, let $\Box\_T$ denote the formalized provability predicate for $T$, and $T\res n$ the theory axiomatized by the axioms of $T$ with Gödel number below $n$.



> 
> **Proposition.** Let $S$ be an extension of $I\Delta\_0+\mathrm{EXP}$, and $T$ be a recursively axiomatized theory such that
>  $$\tag{$\*$}S\vdash\Box\_{T\res n}\phi\implies T\vdash\phi$$
>  for all $n\in\N$ and all $T$-sentences $\phi$. Then every consistent r.e. extension of $T$ is interpretable in a consistent finite extension of $S$.
> 
> 
> 


Observe that $(\*)$ holds whenever $S\subseteq T$ (or just $S$ is $\Sigma\_1$-conservative over $T$) and $T$ is a locally essentially reflexive theory, for example $S=\mathrm{ZC+rank}$ and $T=\mathrm{ZFC}$. But in fact, $(\*)$ holds whenever $S$ is $\Sigma\_1$-sound, which pretty much any natural theory is. Also, it is not difficult to show that if $S$ is locally essentially reflexive, then condition $(\*)$ is *necessary*.


**Proof.** Let $U\supseteq T$ be recursively axiomatized and consistent. Then $(\*)$ implies that
$$V=S+\{\mathrm{Con}\_{U\res n}:n\in\N\}\equiv S+\{\neg\Box\_{T\res n}\neg\phi:n\in\N,\phi\in U\}$$
is consistent; since it is an extension of $S$ by a r.e. set of $\Pi\_1$-sentences, there exists a $\Pi\_1$-sentence $\psi$ such that $S+\psi$ is a $\Sigma\_1$-conservative extension of $V$ by a theorem of Lindström [1]. In particular, $S+\psi$ is consistent, and since it proves $\mathrm{Con}\_{U\res n}$ for all $n\in\N$, it interprets $U$ by the usual interpretation existence lemma. QED


**Reference:**


[1] Per Lindström: *On partially conservative sentences and interpretability*, Proceedings of the American Mathematical Society 91 (1984), no. 3, pp. 436–443, doi: [10.2307/2045318](https://doi.org/10.2307/2045318).


