A strong equivalence between two theories where each is interpretable in the other and the two interpretations compose to interpretations definably isomorphic to the identity; informally, the theories recover each other's structure up to definable isomorphism.

Principle

Requires a pair of translations I: T→S and J: S→T such that the composition J∘I interprets T in a way that is definably isomorphic to the identity interpretation of T (and dually for I∘J), ensuring mutual recoverability of definable structure and metatheoretic properties.

Demonstration

If theory T is interpreted in S via I and S in T via J, and one can define in T the isomorphism witnessing that J∘I acts like the identity on definable objects of T, then T and S are bi‑interpretable; this often appears in examples where different formalisms present 'the same' mathematical structure.

Misapplication

Calling any pair of mutually interpretable theories bi‑interpretable without checking that the compositions are definably isomorphic to identities; confusing bi‑interpretability with weaker equivalences like mutual conservativity or Morita equivalence is a common error.

Consequence

Bi‑interpretability implies a tight matching of definable sets, automorphism groups and many model‑theoretic invariants; it often yields that the theories have the same effective and proof‑theoretic properties up to definable translation.

Reversal

The negation of bi‑interpretability is ordinary asymmetry of interpretability: two theories may interpret each other in different, non‑invertible ways, so that one encodes additional structure not recoverable by the other.

Boundary

Defined for formal theories with suitable definability frameworks; it excludes mere categorical equivalences of model categories or syntactic coincidences that do not provide definable isomorphisms of the composed interpretations.

Semantic Tension

Tension exists between bi‑interpretability and definitional or categorical equivalence: bi‑interpretability is stronger than many pragmatic notions of 'same theory' but weaker than literal syntactic identity, leading to debates about when theories should be considered genuinely the same.

Synthesis

Bi‑interpretability is the condition that two theories encode precisely the same definable structure in mutually recoverable ways: each interprets the other and the round‑trip translations amount to definable identity, establishing a robust equivalence of theories.