Bi-interpretability

Bi-interpretability
Name	Bi-interpretability
Field	Mathematical logic
Related	Model theory, Recursion theory, Proof theory
Notable people	Alfred Tarski, Kurt Gödel, Alfred Tarski, Alfred Tarski, Yuri Matiyasevich

Bi-interpretability Bi-interpretability is a logical relation between formal theories that captures mutual interpretability in a way that yields a strong equivalence of expressive and structural content. It refines notions from Model theory and Mathematical logic to compare theories such as Peano arithmetic, Zermelo–Fraenkel set theory, Second-order arithmetic, and theories arising in Computability theory, Set theory, and Category theory. Bi-interpretability connects with results of Kurt Gödel, Alfred Tarski, Alonzo Church, Alan Turing, and later contributors like Saul Kripke, Harvey Friedman, George Boolos, and Anil Nerode.

Definition and basic concepts

Bi-interpretability is defined using interpretations between formal systems: an interpretation of a theory T in a theory S provides a translation of vocabulary, axioms, and proofs enabling S to simulate T; conversely, T can interpret S. Key earlier concepts include Interpretability logic, Conservative extension, Definitional equivalence, and Mutual interpretability. Definitions use tools from First-order logic, Second-order logic, Infinitary logic, and notions developed by Alfred Tarski, Kurt Gödel, and Alonzo Church. Important surrounding notions include Elementary equivalence, Back-and-forth method, Isomorphism of models, and Categoricity.

Historical development and key contributors

Roots trace to foundational work by David Hilbert, Bertrand Russell, Alfred North Whitehead, and the formalization efforts in Principia Mathematica. Kurt Gödel's incompleteness theorems and Alonzo Church's lambda calculus shaped interpretability notions. Alfred Tarski introduced definability criteria used in later interpretability theory. Mid-20th-century contributors include Jerzy Łoś, Saunders Mac Lane, Alfred Tarski (again), Per Lindström, and Dana Scott. Late 20th-century and contemporary work involves Harvey Friedman, George Boolos, Saul Kripke, Timothy G. Griffin, Richard Montague, Anil Nerode, Jack Silver, Haim Gaifman, Leo Harrington, Jeffrey Paris, Albert Visser, and Vladimir Voevodsky. Developments connect to results and events such as the Hilbert–Bernays provability conditions, the Gödel–Rosser theorem, and work presented at meetings of the Association for Symbolic Logic.

Formal definitions and variants

Formally, an interpretation maps symbols of one signature into definable sets and relations of another structure; bi-interpretability requires existence of interpretations f: T -> S and g: S -> T such that compositions are definably isomorphic to identity interpretations. Variants include weak bi-interpretability, relative bi-interpretability, categorical bi-interpretability, and parameterized forms studied in contexts like Categorical logic, Topos theory, and Homotopy type theory. Technical tools involve Definable isomorphism, Definitional equivalence (as used by Alonzo Church and W. V. O. Quine), and notions from Recursion theory and Descriptive set theory. Related logics include Interpretability logic, Provability logic, and fragments examined by S. Feferman and Solomon Feferman.

Examples and non-examples

Canonical examples: certain expansions of Robinson arithmetic and fragments of Peano arithmetic can be bi-interpretable; Zermelo–Fraenkel set theory with choice relates to class theories and some formulations of Morse–Kelley set theory via strong interpretability relationships. Non-examples include many mutually interpretable but not bi-interpretable theories, such as some pairs of theories arising in Recursively enumerable degrees or incompatible presentations of Second-order arithmetic and weak set theories studied by Simpson and Friedman. Concrete cases involve comparisons among Presburger arithmetic, Peano arithmetic, Robinson arithmetic, and fragments like I Σ1 and I Σn, where bi-interpretability fails or holds depending on definability of coding and truth predicates studied by Tarski, Feferman, Alfred Tarski, and Stephen Kleene.

Properties and preservation results

Bi-interpretability preserves many structural properties: completeness, consistency strength, decidability features, Turing degrees of presentations, and categories of definable sets. Preservation results relate to invariants studied by Svenonius, Back-and-forth arguments used by Alfred Tarski and Marshall Hall Jr., and criteria from Beth definability theorem and Craig's interpolation theorem with contributors like William Craig. Connections appear with Automorphism groups of models studied by Évariste Galois-inspired approaches in model theory and results by Hodges and Marker. Failures of preservation lead to counterexamples constructed by Harvey Friedman, Leo Harrington, and Saharon Shelah.

Applications in model theory and logic

Bi-interpretability serves to transfer theorems between theories in Model theory and to classify theories up to strong equivalence; it is used in work on Categoricity, Stability theory, and structural analysis of arithmetic and set theory. Applications include reductions in Reverse mathematics (linked to Stephen Simpson), analysis of definability in Algebraic geometry contexts connected to Grothendieck-inspired foundations, and in comparisons of formalizations used in Proof assistants such as Coq, Lean and Isabelle. It influences computability-theoretic classifications by researchers like Richard Shore and Andrea Sorbi and underpins philosophical discussions by Hilary Putnam, Wilfrid Sellars, and Michael Dummett.

Open problems and research directions

Open problems include classification of natural theories up to bi-interpretability, decidability of bi-interpretability for given classes, connections with categorical equivalences in Higher category theory and Homotopy type theory as explored by Vladimir Voevodsky and Jacob Lurie, and interactions with Descriptive set theory invariants from research by Kechris and Friedman. Ongoing directions study bi-interpretability in computational settings (links to Complexity theory figures like Cook and Ladner), extensions to infinitary logics studied by Per Lindström, and applications to mechanized mathematics in Proof theory initiatives led by Georges Gonthier and Jeremy Avigad.

Category:Mathematical logic