Infinitary logic
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Infinitary logic is the study of mathematical languages which permit expressions of infinite size. In most infinite logics expressions are well founded trees of symbols. Proofs in infinite logic are well ordered sequences of expressions each of which is an axiom, follows from two previous statements by modus ponens or is a conjunction of previous statements. Most infinite logics put a bound on the cardinality of quantifiers used and the length of statements.
Most infinite languages fail to be complete or compact. The question as to whether a certain infinite logic named <math>\Omega<math> logic is complete promises to through light on the continuum hypothesis.
Certain statements that cannot be expressed in finite languages such as the concept of well foundedness are trivially expressed in various infinite logics.