Noninfinitary

Noninfinitary refers to systems, languages, or methods in logic and mathematics that do not employ infinitary operations. In this context, all expressions, proofs, and syntactic constructions are finite in length. The term is used to contrast finitary or standard approaches with infinitary logics, which permit infinite conjunctions, disjunctions, or quantifier structures.

In logic, the prototypical noninfinitary framework is first-order logic (FOL), along with propositional logic. FOL formulas are finite strings built from variables, predicate symbols, function symbols, logical connectives, and finite quantifier prefixes. By contrast, infinitary logics, such as L_{κ,λ}, allow conjunctions and disjunctions over sets of formulas of size up to κ and quantifier blocks of length up to λ, which can be infinite.

The choice of noninfinitary systems supports important meta-theoretic properties. For example, first-order logic enjoys the completeness and compactness theorems, and a variety of models and theories can be analyzed using well-developed finitary techniques. However, finitary frameworks can be limited in expressive power, unable to capture certain concepts that require infinite constructions. Infinitary logics address some of these gaps but at the cost of more complex semantics and weaker or different model-theoretic results.

Applications of noninfinitary formalisms span mathematics, computer science, and formal reasoning. They underpin everyday formalizations, programming language semantics, and many automated theorem-proving and verification methods, where finite syntax and finite proofs are essential.

See also: infinitary logic, first-order logic, finite model theory, proof theory.