An
infinitary logic is a
logic that allows infinitely long
statements and/or infinitely long
proofs. Some infinitary logics may have different properties from those of standard
first-order logic. In particular, infinitary logics may fail to be
compact or
complete. Notions of compactness and completeness that are equivalent in
finitary logic sometimes are not so in infinitary logics. Therefore for infinitary logics, notions of strong compactness and strong completeness are defined. This article addresses Hilbert-type infinitary logics, as these have been extensively studied and constitute the most straightforward extensions of finitary logic. These are not, however, the only infinitary logics that have been formulated or studied.