A language counting formulae of infinite length as members. The two dimensions frequently encountered are infinitary conjunctions and disjunctions and infinitary strings of quantifiers. In the former case, conjunction and disjunction over an infinite set of formulae $\Phi$ is defined as

• • $\bigwedge \Phi$

• • $\bigvee \Phi$

respectively. Infinite strings of universal and existential quantifiers are handled similarly. Typically, where $ℒ$ is a finitary first-order language, its infinitary generalizations are described by ${ℒ}_{\kappa \lambda }$ where the cardinals $\kappa$ and $\lambda$ indicate that conjunctions and disjunctions of length less than $\kappa$ and strings of quantifiers of length less than $\lambda$ are well-formed. For example, the finitary language of classical logic is ${ℒ}_{\omega \omega }$, as, e.g., conjunctions must be smaller than $\omega$, that is, finite (by convention, $\omega$ is used rather than ${\aleph }_0$).