WHEN BI-INTERPRETABILITY IMPLIES SYNONYMY

This paper studies a central question in model theory and the philosophy of mathematics: when are two formal theories “the same”? Two prominent notions of sameness are (i) synonymy (also called definitional equivalence), which is the strictest common notion, and (ii) bi-interpretability, which is weaker and allows the theories to interpret each other up to definable isomorphism rather than literal definitional extension.

The authors’ research question is: in what circumstances can one infer synonymy from bi-interpretability? This matters because bi-interpretability is often easier to establish and is known to preserve many structural properties (e.g., automorphism groups up to isomorphism, categoricity-like properties, finite axiomatizability). Synonymy, however, preserves more fine-grained structure—most notably, it preserves the action of automorphisms on the domain (not just the abstract automorphism group). Thus, knowing when bi-interpretability collapses to synonymy tells us when “mutual interpretability” is strong enough to guarantee genuine definitional equivalence.

The paper focuses on a special class of theories called sequential theories. Sequentiality is motivated by the presence of coding of sequences (and more generally, enough internal set-like structure to build sequences and truth predicates). Examples given include Buss’s theory ${\mathrm{S}}_2^1$, elementary arithmetic (EA/EFA), $\mathrm{I}{\Sigma }_1$, and set theories such as ZF and ZFC. The authors also introduce a broader notion, conceptuality, which is proof-generated and is slightly more general than sequentiality; all sequential theories are conceptual, but not conversely.

Methodologically, the paper is not empirical but proof-theoretic: it develops categorical and model-theoretic machinery around interpretations, then proves a key theorem using an internal version of the Schröder–Bernstein theorem.

The core definitions are: an interpretation $K:U\to V$ is given by a translation of the language of $U$ into $V$, with a domain formula $\delta (\vec{x})$ (of some dimension $m$) specifying which tuples in models of $V$ serve as the interpreted domain. Interpretations can be composed, and the paper distinguishes several “dimensions” and properties: one-dimensional interpretations, identity-preserving interpretations (translating identity to identity), unrelativized/direct interpretations, etc. Synonymy corresponds to isomorphism in a category ${\mathsf{INT}}_0$ of interpretations modulo “equality” (the target theory proves the interpretations coincide). Bi-interpretability corresponds to isomorphism in ${\mathsf{INT}}_1$, where the target theory provides definable isomorphisms between the internal models produced by compositions.

The main technical ingredient is a version of the Schröder–Bernstein theorem under very weak conditions. The authors formalize this in an auxiliary theory $\mathsf{SB}$, built from adjunctive class theory $\mathsf{ac}$ plus axioms ensuring the existence of two equivalence relations and injections between quotient classes. Concretely, $\mathsf{SB}$ has unary predicates $A$, $B$ and binary predicates ${\mathsf{E}}_A$, ${\mathsf{E}}_B$, plus injections $F$ and $G$ between the quotient sets $A/{\mathsf{E}}_A$ and $B/{\mathsf{E}}_B$. The proof constructs a definable bijection $H$ between these quotients using a “switch” argument: for each element $x$, either there is an $x$-switch (a certain downward-closed pair of virtual classes) or not, and the definition of $H$ chooses between $F$ and $G$ accordingly. The paper then proves within $\mathsf{SB}$ that $H$ is well-defined on equivalence classes, functional, injective, and surjective.

With this Schröder–Bernstein machinery, the main theorem is proved as follows. Suppose two theories $U$ and $V$ are bi-interpretable via interpretations $K:U\to V$ and $M:V\to U$. The authors assume that the interpretations involved in the bi-interpretation are one-dimensional and identity-preserving, and that $V$ is conceptual (in particular, sequential). Under these assumptions, they show that synonymy follows.

The key conceptual step is that the bi-interpretation gives mutual “retract-like” behavior: $V$ proves that the composition $K\circ M$ yields the identity interpretation on $V$ (in the appropriate sense), while $U$ proves that $M\circ K$ yields an interpretation elementarily equivalent to the identity on $U$. To upgrade from bi-interpretability to synonymy, they construct a direct, identity-preserving interpretation $K^{\prime }$ by using Schröder–Bernstein to turn definable injections between the full domain and the interpreted domain into a definable bijection. This allows them to replace the original interpretation by a direct one that is isomorphic to it, and then apply a corollary stating that if one witnessing interpretation is direct (and the other conditions hold), then synonymy results.

The paper also addresses optimality. It provides an example showing the theorem cannot be strengthened by dropping identity preservation for both interpretations: there exist two finitely axiomatized sequential theories that are bi-interpretable but not synonymous, where precisely one of the interpretations in the bi-interpretation is not identity-preserving. The example is framed as “Frege meets Cantor”: the authors define a one-sorted weak set theory ${\mathsf{ACF}}_{♭}$ (adjunctive class theory with Frege relation) and compare it to adjunctive set theory $\mathsf{AS}$. They give explicit one-dimensional interpretations witnessing bi-interpretability between $\mathsf{AS}$ and ${\mathsf{ACF}}_{♭}$, with one interpretation identity-preserving. They then show non-synonymy by analyzing automorphism actions on domains. Specifically, they construct a model $\mathcal{M}$ of $\mathsf{AS}$ with an automorphism $\sigma$ of order 2 that fixes any given finite set but has only finitely many fixed points overall. If synonymy held, the internal model of ${\mathsf{ACF}}_{♭}$ inside $\mathcal{M}$ would be definable using only finitely many parameters $X_0$, so $\sigma$ would induce an automorphism of the internal model. They then consider classes of the form $\{p,\sigma p\}$ and argue that there would be infinitely many such classes fixed by $\sigma$ by extensionality, contradicting that $\sigma$ has only finitely many fixed points. This shows the separation between bi-interpretability and synonymy is real and tied to the identity-preserving requirement.

Limitations are primarily conceptual rather than empirical: the theorem applies to conceptual (hence sequential) theories and requires one-dimensional, identity-preserving interpretations. The authors explicitly argue optimality by giving a counterexample when identity preservation is not assumed for one of the interpretations. They also note that sequentiality is not closed under bi-interpretability in general, motivating the use of the broader conceptuality framework.

Practical implications: while the results are theoretical, they guide how logicians should interpret “sameness” claims. If you can prove bi-interpretability between sequential/conceptual theories using identity-preserving one-dimensional interpretations, then you can safely conclude synonymy, meaning definitional equivalence and preservation of domain-level automorphism action. This is important for transferring results between theories: definitional equivalence is stronger than mutual interpretability and supports more faithful translation of structures and invariants.

Who should care? Model theorists and logicians studying interpretability, definitional equivalence, and the structure of theories under translations; philosophers of mathematics concerned with what it means for theories to be “the same”; and researchers in proof theory and foundational logic who use interpretability as a tool for comparing systems (e.g., arithmetic and set theories). The paper also contributes an independently interesting Schröder–Bernstein theorem formalized under weak assumptions, which may be useful in other interpretability arguments.

Overall, the paper’s core contribution is a sharp criterion: under conceptuality and identity-preserving one-dimensional bi-interpretations, bi-interpretability collapses to synonymy, with a matching example showing the criterion is essentially tight.