Heyting arithmetic¶
A first-order theory of natural-number arithmetic using intuitionistic rather than classical logic while retaining arithmetic axioms and induction.
Core Idea¶
Heyting arithmetic rejects unrestricted excluded middle and double-negation elimination, admits constructive interpretations and realizability, and is equiconsistent with Peano arithmetic while differing in theoremhood and proof meaning. Natural-number terms and arithmetic axioms are combined with intuitionistic predicate-calculus rules; induction constructs propositions over numbers, and translations relate classical proofs to intuitionistic ones without identifying their inferential regimes. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Heyting arithmetic belongs to intuitionistic logic and foundations of arithmetic and is useful where the analyst can specify the typed intuitionistic logic and foundations of arithmetic carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the first-order signature, equality and recursive function symbols, intuitionistic inference rules, induction schema, treatment of falsity and negation, excluded-middle status, intended or proof-theoretic semantics, translation theorem, and comparison with Peano arithmetic are explicit. The scope is broad within that domain but bounded by the need for the first-order signature, equality and recursive function symbols, intuitionistic inference rules, induction schema, treatment of falsity and negation, excluded-middle status, intended or proof-theoretic semantics, translation theorem, and comparison with Peano arithmetic are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the first-order signature, equality and recursive function symbols, intuitionistic inference rules, induction schema, treatment of falsity and negation, excluded-middle status, intended or proof-theoretic semantics, translation theorem, and comparison with Peano arithmetic are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Heyting arithmetic. Heyting arithmetic compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed intuitionistic logic and foundations of arithmetic carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of intuitionistic logic and foundations of arithmetic because they reuse the typed intuitionistic logic and foundations of arithmetic carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Natural-number terms and arithmetic axioms are combined with intuitionistic predicate-calculus rules; induction constructs propositions over numbers, and translations relate classical proofs to intuitionistic ones without identifying their inferential regimes., and type the carrier, state every parameter and convention in the definition, test that the first-order signature, equality and recursive function symbols, intuitionistic inference rules, induction schema, treatment of falsity and negation, excluded-middle status, intended or proof-theoretic semantics, translation theorem, and comparison with Peano arithmetic are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Heyting arithmetic Domain-specific
Parents (1) — more general patterns this builds on
-
Heyting arithmetic is a kind of Deductive Reasoning Prime
The proposed strict upward parent is
prime:deductive_reasoning.
Hierarchy path (1) — routes to 1 parentless root
- Heyting arithmetic → Deductive Reasoning
Neighborhood in Abstraction Space¶
Heyting arithmetic sits in a crowded region of the domain-specific corpus (39th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Set Theory & Constructive Foundations (15 abstractions)
Nearest neighbors
- Bounded arithmetic — 0.91
- Monadic predicate calculus — 0.90
- Finite set — 0.89
- Self-verifying theories — 0.89
- Benacerraf's identification problem — 0.89
Computed from structural-signature embeddings · 2026-09-08