Infinitary Logics
An infinitary logic is a Formal logical system, logic that allows infinitely long statement (logic), statements and/or infinitely long Mathematical proof, proofs. The concept was introduced by Zermelo in the 1930s. Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be Compactness (logic), compact or Completeness (logic), 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 system, 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. Considering whether a certain infinitary logic named Ω-logic is complete promises t ...
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