Non-classical logic | Systems of formal logic

Infinitary logic

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. Considering whether a certain infinitary logic named Ω-logic is complete promises to throw light on the continuum hypothesis. (Wikipedia).

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Related pages

Completeness (logic) | Regular cardinal | Hilbert system | Axiom of regularity | Sentence (mathematical logic) | Set theory | Theory (mathematical logic) | Ω-logic | Mathematical proof | Continuum hypothesis | Logical disjunction | Weakly compact cardinal | Cardinality | Sequence | Craig interpolation | First-order logic | Strongly compact cardinal