Heyting Arithmetic (HA) Overview

• Heyting Arithmetic (HA) is the foundational first-order arithmetic theory based on intuitionistic logic, emphasizing constructive proofs over classical reasoning.
• It employs a formal language with standard arithmetic operations, induction axioms, and stratified logical principles that calibrate constructive mathematics and modal provability logics.
• HA's extensions include higher-order systems, fixpoint theories, and cyclic proof systems, linking it to type theory, constructive set theory, and advanced proof analysis.

Heyting Arithmetic (HA) is the foundational first-order theory of arithmetic based on intuitionistic logic rather than classical logic. Its significance arises from both its independent interest in mathematical logic and the fact that its principles, variant subsystems, admissible inferences, and proof-theoretic extensions provide precise calibration points for constructive mathematics, type theory, and modal provability logics.

1. Formal Description and Logical Foundations

Heyting Arithmetic is formulated in the first-order language $\mathcal{L}_{AR} = \{0, S, +, \cdot, =\}$, where $0$ is the constant symbol for zero, $S$ the successor function, and $+$, $\cdot$, and $=$ standard addition, multiplication, and equality. The logical basis is intuitionistic first-order predicate logic with equality ($\land, \lor, \to, \forall, \exists$), omitting the law of excluded middle.

The axioms of HA comprise:

• Arithmetic axioms:

1. $S(x) \neq 0$
2. $S(x) = S(y) \to x = y$
3. $x+0 = x$, $0$0
4. $0$1, $0$2

• Induction scheme: For every formula $0$3,

$0$4

If the law of excluded middle were added for all formulas, HA would collapse to classical Peano Arithmetic (PA), but constructive proofs and intuitionistic reasoning exhibit crucial logical differences, especially for statements with high quantifier complexity (Olsson et al., 2021).

2. Proof-theoretic Properties and Hierarchies

HA admits a fine stratification of logical and arithmetical principles by quantifier complexity. Key hierarchies include:

• Restricted Law of Excluded Middle ($0$5, $0$6): For $0$7, $0$8-formulas, LEM can be adopted in a restricted manner. Over HA, for each $0$9:
  • $S$0
  • Ascending the hierarchy yields ever stronger systems, not collapsible in general, and essential for calibrating the passage from constructive to classical arithmetic (Fujiwara et al., 2020).
• Inductive Fragments: Restricting induction to formulas in bounded complexity classes yields the $S$1 theories; these form an intricate web of subtheories critical in proof analysis and constructive reverse mathematics (Jeon, 2020).

3. Admissibility, Unification, and Provability Logic

The characterization of HA's admissible rules and substitutions is sharpened using projective resolutions and the class of NNIL (No Nested Implications on the Left) formulas. The provability logic of HA is axiomatized by a decidable system that accounts for how $S$2-sentences are provable, employing Ghilardi's projective unifiers and Visser–Montagna rules explicitly (Mojtahedi, 2022).

The system $S$3 extends intuitionistic modal logic with box ($S$4) and captures the $S$5-provability logic of HA (and, more precisely, of its "completion" $S$6, see Section 5). Its axioms include the K4, Löb, Completeness Principle (CP, $S$7), and a family of schematic axioms designed to encode admissible inferences at the propositional level (Ardeshir et al., 2018).

4. Extensions: Higher-order, Fixpoints, and Type Theory

Various proof-theoretic and semantic extensions of HA underscore its central role:

• Higher-order Heyting Arithmetic ($S$8 and HAH): By extending the types to all finite (simple) types, $S$9 becomes the formal core for system T, type-theoretic and functional interpretations. Key theorems, such as the Bar Theorem, show that every term-definable function $+$0 is uniformly continuous on the Cantor space and admits a bar induction principle (Kawai, 2019).

In modern type theory, universe-free dependent type theory is conservative over HA, and adding one impredicative universe gives full higher-order HA (HAH), but with no new first-order consequences. Different logical translations (proof-relevant vs. proof-irrelevant) precisely locate the strength of $+$1-calculi and type theories vis-à-vis HA and HAH (Berg et al., 2023).

• Fixpoints ($+$2): Adding predicates for strictly positive fixed points to HA, even with full impredicative comprehension, does not yield new first-order arithmetic theorems. Detailed realizability arguments using partial terms and satisfaction-predicate hierarchies demonstrate that the extended theory is conservative (Olsson et al., 2021).

5. Bi-interpretability and Constructive Set Theory

A constructive analogue of Ackermann’s theorem is established: HA is bi-interpretable with $+$3, the finitary version of Constructive Zermelo–Fraenkel set theory. Interpreting arithmetic into set theory and vice versa via bit-codings or ordinal representations, it is shown that constructive analysis of finiteness, induction, and bounding principles produce mutual conservative translations between the two foundations (Jeon, 2020).

Class Formula Hierarchies in HA↔CZF^{fin}: | Class | HA fragment | CZF^{fin} fragment | |-------|-------------|--------------------| | $+$4 | $+$5 (induction for $+$6-formulas) | $+$7 (Set-induction for $+$8) |

6. Modal and Epistemic Connections

Via the Gödel–Rasiowa–Sikorski "box" translation and its Flagg–Friedman refinement, HA embeds faithfully into epistemic arithmetic (EA: PA plus S4 modalities). The modal disjunction property and numerical existence property in EA correspond, via such embeddings, to the classical disjunction property (DP) and numerical existence property (NEP) in HA. Notably, Markov's Rule in HA is shown to be equivalent to its epistemic form in EA, with further strengthening in modal-epistemic formalisms (Inoué, 2023).

Key results include:

• If EA satisfies modal DP, then HA satisfies DP.
• The box ($+$9) and modified RS translations preserve provability and admit faithfulness of embedding in both directions.

7. Cyclic and Non-classical Proof Systems

Recent research investigates cyclic proof systems for HA, which replace induction axioms with finite cyclic graphs in the sequent calculus. The class $\cdot$0 (cyclic proofs with cycles on $\cdot$1-formulas) is shown to embed in the next higher inductive fragment $\cdot$2, providing new perspectives on the computational content and automation of inductive reasoning (Leigh et al., 28 Jul 2025).

Features:

• Soundness and completeness relative to standard HA.
• Translation mechanisms that extract a global "induction invariant" from cyclic graphs, enabling reduction to ordinary finitary induction proofs.

• "The $\cdot$3-Provability Logic of HA*" (Ardeshir et al., 2018)
• "Relative Unification in Intuitionistic Logic: Towards provability logic of HA" (Mojtahedi, 2022)
• "Revisiting the conservativity of fixpoints over intuitionistic arithmetic" (Olsson et al., 2021)
• "Constructive Ackermann's interpretation" (Jeon, 2020)
• "Unravelling Cyclic First-Order Arithmetic" (Leigh et al., 28 Jul 2025)
• "Representing definable functions of $\cdot$4 by neighbourhood functions" (Kawai, 2019)
• "Epistemic systems and Flagg and Friedman's translation" (Inoué, 2023)
• "Refining the arithmetical hierarchy of classical principles" (Fujiwara et al., 2020)
• "Conservativity of Type Theory over Higher-order Arithmetic" (Berg et al., 2023)
• "Strong Normalization for HA + EM1 by Non-Deterministic Choice" (Aschieri, 2013)
• "Interactive Learning-Based Realizability for Heyting Arithmetic with EM1" (Aschieri et al., 2010)