Infinitary logics

An infinitary logic arises from ordinary first-order logic when one or more of its finitary properties is allowed to become infinite, for example, by admitting infinitely long formulas or infinitely long or infinitely branched proof figures. The need to extend first-order logic became pressing in the late 1950s when it was realized that many of the fundamental notions of mathematics cannot be expressed in first-order logic in a way that would allow for their logical analysis. Because infinitary logics often do not suffer the same limitation, they have become an essential tool in mathematical logic.