Infinitary logic

Computer/Terms 2008. 4. 1. 18:40

An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. Infinitary logics have different properties from those of standard first-order logic. In particular, infinitary logics often fail to be compact or complete. Notions of compactness and completeness that are equivalent in finitary logic sometimes are not so in infinitary logic. So for infinitary logics the notions of strong compactness and strong completeness are defined. In this article we shall be concerned with 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 around.

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

Reference:
http://en.wikipedia.org/wiki/Infinitary_logic

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