aritmetikailogikai

Aritmetikailogikai is a field of mathematical logic that studies formal systems in which the arithmetic of natural numbers is expressed and analyzed within a logical framework. It investigates how arithmetic can be captured by axioms, rules of inference, and formal languages, and how the strength of different axiomatizations affects provability and truth.

The central objects are axiom systems for arithmetic, such as Peano arithmetic, Robinson arithmetic, and Presburger arithmetic, as well as extensions to include multiplication, higher-order formulations, and alternative logical foundations. Peano arithmetic (PA) is a standard first-order theory of the natural numbers with induction; Presburger arithmetic handles addition (and order) and is decidable; Robinson arithmetic (Q) is a weaker, primarily universal fragment. Researchers study consistency, relative strength, interpretability, and the expressive power of these theories.

A core line of results concerns decidability and incompleteness. Gödel’s incompleteness theorems show that any consistent, sufficiently strong axiomatization of arithmetic cannot prove all arithmetical truths and cannot verify its own consistency. The field also examines decidability boundaries, the arithmetical hierarchy, and the connections between arithmetic and computability through recursion theory.

Methodologically, aritmetikailogikai employs proof theory, model theory, and computability to analyze theories, construct models, and compare different formal systems. It contributes to the foundations of mathematics and informs areas such as automated theorem proving and formal verification.

See also: mathematical logic, arithmetic, Gödel, Hilbert program, Peano axioms.