Finitary

In mathematics or logic, a finitary operation is an operation that takes a finite number of input values to produce an output, like those of arithmetic. Operations on infinite numbers of input values are called infinitary.

Finitary argument

A finitary argument is one which can be translated into a finite set of symbolic propositions starting from a finite set of axioms. In other words, it is a proof (including all assumptions) that can be written on a large enough sheet of paper.

By contrast, infinitary logic studies logics that allow infinitely long statements and proofs. In such a logic, one can regard the existential quantifier, for instance, as derived from an infinitary disjunction.

History

The emphasis on finitary methods has historical roots.

In the early 20th century, logicians aimed to solve the problem of foundations; that is, answer the question: "What is the true base of mathematics?" The program was to be able to rewrite all mathematics starting using an entirely syntactical language without semantics. In the words of David Hilbert (referring to geometry), "it does not matter if we call the things chairs, tables and beer mugs or points, lines and planes."