infinitary

Infinitary is an adjective used in mathematics, logic, and related fields to indicate notions involving infinity or unlimited size. It commonly describes operations, constructions, languages, or theories that allow infinitely many components.

In logic, infinitary logic extends first-order logic by permitting formulas with infinitely long conjunctions or disjunctions and, in some variants, infinite blocks of quantifiers. The standard notation is L_{κ,λ}, where κ bounds the maximum length of conjunctions or disjunctions and λ bounds the number of distinct variables or quantifiers that may appear in a formula. For instance, L_{ω,ω} is the ordinary finitary first-order logic; L_{ω1,ω} allows countable conjunctions and disjunctions with a countable vocabulary. Infinitary logics are powerful tools for analyzing properties that escape finitary description, but they typically lose some meta-theoretic properties such as compactness and, in general, completeness. They play a central role in model theory, including investigations of categoricity, spectra of theories, and the behavior of models under infinite constructions.

In set theory and infinitary combinatorics, infinitary methods study infinite structures and relations, such as partition relations for infinite cardinals and Ramsey theory beyond finite sets. These approaches interact with large cardinal axioms and descriptive set theory, providing a framework for understanding the limits of definability and regularity properties of infinite objects.