Elementary function arithmetic

A weak first-order arithmetic theory whose provably total functions are the elementary recursive functions, typically extending bounded arithmetic with exponentiation.

Version

v1 · 2026-09-08 · History

Domain-specific #

4341

Origin domain

proof theory

Subdomain

proof theory

Core Idea

Elementary function arithmetic formalizes arithmetic with symbols or axioms sufficient for addition, multiplication, exponentiation, and induction restricted to bounded formulas. Bounded quantification and restricted induction permit elementary towers of fixed height while excluding stronger growth and many proofs available in Peano arithmetic. 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.

The load-bearing residual is not the broad topic of proof theory. It is the domain-specific identity determined by the language, exponentiation axioms, bounded-induction scheme, and resulting elementary provably total functions match the declared EFA presentation.

Scope of Application

Elementary function arithmetic belongs to proof theory and is useful where the analyst can specify the typed proof theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the language, exponentiation axioms, bounded-induction scheme, and resulting elementary provably total functions match the declared EFA presentation. The scope is broad within that domain but bounded by the need for the language, exponentiation axioms, bounded-induction scheme, and resulting elementary provably total functions match the declared EFA presentation. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity

The abstraction clarifies a crowded vocabulary by making the language, exponentiation axioms, bounded-induction scheme, and resulting elementary provably total functions match the declared EFA presentation the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Elementary function arithmetic can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

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 Elementary function arithmetic. Elementary function 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

1. Identify the carrier. State what the elements, states, objects, or observations are: the typed proof theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the language, exponentiation axioms, bounded-induction scheme, and resulting elementary provably total functions match the declared EFA presentation independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of proof theory because they reuse the typed proof theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Bounded quantification and restricted induction permit elementary towers of fixed height while excluding stronger growth and many proofs available in Peano arithmetic., and type the carrier, state every parameter and convention in the definition, test that the language, exponentiation axioms, bounded-induction scheme, and resulting elementary provably total functions match the declared EFA presentation, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Elementary function arithmetic Domain-specific

Parents (1) — more general patterns this builds on

• Elementary function arithmetic is a kind of Boundedness Prime

  The proposed strict upward parent is prime:boundedness.

Hierarchy path (1) — routes to 1 parentless root

• Elementary function arithmetic → Boundedness

Neighborhood in Abstraction Space

Elementary function arithmetic sits in a moderately populated region (44^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Algebraic Number Theory & Reciprocity (28 abstractions)

Nearest neighbors

• Square number — 0.90
• Bounded arithmetic — 0.89
• Geometric progression — 0.89
• Arithmetic function — 0.89
• Gauss's lemma (number theory) — 0.89

Computed from structural-signature embeddings · 2026-09-08