Heyting arithmetic

Heyting arithmetic
Name	Heyting arithmetic
Alternative names	Intuitionistic Peano arithmetic
Discipline	Mathematical logic
Introduced	1930s
Notable figures	Arend Heyting, L. E. J. Brouwer, David Hilbert, Gerhard Gentzen, Kurt Gödel, Andrey Kolmogorov

Heyting arithmetic is a formal system for arithmetic based on intuitionistic logic, serving as an intuitionistic counterpart to classical Peano arithmetic. It formalizes natural number arithmetic using the axioms of Peano axioms while replacing classical logic with the proof interpretations associated with Brouwerian philosophy, L. E. J. Brouwer, and later constructs by Arend Heyting. The system plays a central role in studies connecting constructivism, proof theory, and computability theory.

Overview

Heyting arithmetic is formulated to reflect principles endorsed by L. E. J. Brouwer and developed by Arend Heyting and others; it contrasts with systems influenced by David Hilbert and Hilbert's program. It adopts intuitionistic logic related to the Brouwer fixed-point theorem only insofar as historical philosophical roots, and it has been analyzed in relation to results by Kurt Gödel and Gerhard Gentzen. The study of Heyting arithmetic intersects with work by Andrey Kolmogorov on the interpretation of intuitionistic logic, with connections to Alan Turing and Alonzo Church through computability interpretations.

Formal system

The language of Heyting arithmetic uses the vocabulary familiar from Peano arithmetic, including symbols introduced by Giuseppe Peano and axiomatizations influenced by Richard Dedekind. The axioms encompass successor, zero, addition, multiplication, and induction schemes as in systems studied by Ernst Zermelo and Thoralf Skolem in first-order settings. Logical rules derive from formulations by Gerhard Gentzen and others who articulated sequent calculi and natural deduction systems. Proof techniques in the system are connected historically to methods employed by Emil Post and Stephen Kleene, and modern presentations often reference frameworks elaborated by Per Martin-Löf and Dag Prawitz.

Metamathematical properties

Metamathematical analysis of Heyting arithmetic uses tools from work by Kurt Gödel on incompleteness and by Gerhard Gentzen on consistency proofs. Heyting arithmetic inherits many incompleteness phenomena studied by Gödel, although intuitionistic provability predicates call for adaptations in proofs similar to those by Hilbert and Bernays and later by J.-Y. Girard. Conservativity results involving fragments relate to results by John Myhill, while proof-theoretic strength comparisons reference ordinals and methods developed by Wilhelm Ackermann and Gentzen; connections to ordinal analysis and results by Georg Kreisel are standard. The system's constructive content has been explored via realizability interpretations introduced by Stephen Kleene and extended by Shinichi Mochizuki and Solomon Feferman in various contexts. Consistency relative to classical systems often leverages techniques inspired by Gerhard Gentzen and Kurt Gödel.

Models and semantics

Semantic treatments of Heyting arithmetic include Kripke models developed by Saul Kripke, alongside Beth models and topological models inspired by work of Errett Bishop and others promoting constructive approaches. Modal and categorical semantics relate to contributions by William Lawvere and F. William Lawvere in categorical logic, and to the internal logic of topos theory as developed by Alexander Grothendieck and William Lawvere. Realizability models stem from constructions by Stephen Kleene and have been refined by Dana Scott and Per Martin-Löf, while recursive models connect to classical recursive model theory advanced by S. C. Kleene and G. N. Raney. The interplay between Kripke semantics and categorical semantics links to investigations by Anders Kock and Max Kelly.

Relationships to other systems

Heyting arithmetic relates closely to Peano arithmetic as its intuitionistic analogue, and comparisons often involve conservativity and interpretability results akin to those studied by Gerhard Gentzen and Kurt Gödel. Relationships to type-theoretic systems draw on work by Per Martin-Löf and Alonzo Church in the lambda calculus tradition formalized by Haskell Curry and Robert Harper. Connections to constructive set theories reference developments by Errett Bishop and Myhill and to higher-order arithmetic through analyses by Simpson and collaborators of subsystems studied in reverse mathematics by Stephen G. Simpson. Interpretations via categorical logic and topos-theoretic embeddings point to contributions by William Lawvere and F. W. Lawvere and developments in topos theory initiated by Alexander Grothendieck.

Applications and historical context

Historically, Heyting arithmetic emerged from exchanges among L. E. J. Brouwer, Arend Heyting, David Hilbert, and later commentators such as Kurt Gödel and Gerhard Gentzen. Its applications are prominent in constructive mathematics advocated by Errett Bishop and in proof-theoretic investigations by Georg Kreisel and Gerhard Gentzen. Computational interpretations via realizability link to seminal work by Alan Turing, Alonzo Church, and Stephen Kleene, and modern uses appear in proof assistants influenced by systems developed at institutions like Princeton University, University of Cambridge, and Massachusetts Institute of Technology. Heyting arithmetic continues to inform research programs associated with constructive analysis, type theory research at Institute for Advanced Study, and studies in the foundations of mathematics by scholars such as Solomon Feferman and Michael Dummett.

Category:Mathematical logic