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First-Order Arithmetic

Formal first-order theories of natural-number arithmetic defined by a signature, axioms, models, and a specified induction scheme.

Version
v1 · 2026-09-28 · History
Domain-specific #
9472
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Mathematical Logic, Theories of Arithmetic → Mathematics
Aliases
First-order theories of arithmetic

Core Idea

First-order arithmetic comprises formal theories in a first-order language for natural-number structures, using symbols for zero, successor, arithmetic operations, and relations together with specified axioms and induction strength.

Peano arithmetic uses first-order axioms plus an induction scheme for formulas in its language. Robinson arithmetic keeps basic operation axioms but lacks full induction, illustrating a weaker theory.

Scope of Application

  • Logic. Studies theories and models.
  • Foundations. Formalizes elementary number reasoning.
  • Proof theory. Compares induction strength.
  • Model theory. Analyzes nonstandard arithmetic structures.

Clarity

Include first-order axiom systems intended to formalize natural-number arithmetic with a declared language and induction resources. Exclude elementary-school calculation, second-order Peano arithmetic, set theory used as a foundation, and the complete true arithmetic theory presented as computably axiomatizable. Inclusion test: Include first-order axiom systems intended to formalize natural-number arithmetic with a declared language and induction resources. Exclusion test: Exclude elementary-school calculation, second-order Peano arithmetic, set theory used as a foundation, and the complete true arithmetic theory presented as computably axiomatizable. Nearest boundary: Presburger arithmetic is a first-order arithmetic theory but omits multiplication as a full operation and has different strength. Exit condition: The category changes when quantifiers range over sets or the intended subject is not number arithmetic. Common misclassifications: It is not ordinary arithmetic practice. It is not automatically Peano arithmetic. It is not second-order arithmetic. It is not identical to all truths of the standard natural numbers. Nearest named distinctions: Peano arithmetic: One major theory in the family. True arithmetic: All first-order truths of the standard model. Presburger arithmetic: Addition-only decidable arithmetic. Second-order arithmetic: Allows set-of-number variables.

Manages Complexity

First-order axioms target natural numbers yet admit other models. Stronger induction proves more but cannot capture all arithmetic truth effectively.

Abstract Reasoning

  1. Signature — Lists zero, successor, operations, and relations. Different signatures express different formulas.
  2. Formulas — Build statements with first-order quantifiers. Second-order set quantification changes the logic.
  3. Axioms — Constrain intended arithmetic behavior. A language alone is not a theory.
  4. Models — Interpret symbols in structures satisfying axioms. Nonstandard models matter in first-order theories.
  5. Induction scheme — Controls which formulas support induction. Removing or restricting it changes proof power.
  6. Proof relation — Derives theorems by first-order rules. Truth in the standard model and provability are distinct.

Knowledge Transfer

Signature–axiom–model analysis transfers across formal theories, but induction strength, decidability, and standard-model claims must be reproved for each arithmetic language.

Relationships to Other Abstractions

Local relationship map for First-Order ArithmeticParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.First-OrderArithmeticDOMAINDomain-specific abstraction: Formal Theory — is a kind ofFormal TheoryDOMAIN

Current abstraction First-Order Arithmetic Domain-specific

Parents (1) — more general patterns this builds on

  • First-Order Arithmetic is a kind of Formal Theory Domain-specific

    First-Order Arithmetic is a strict kind of Formal Theory: it is a first-order theory fixed by arithmetic signature, axioms, models, and proof system.

Hierarchy paths (2) — routes to 2 parentless roots

Neighborhood in Abstraction Space

First-Order Arithmetic sits in a crowded region of the domain-specific corpus (31st percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Logical Connectives & Formal Systems (13 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08