Шексіздік логикасы: қасиеттері мен түрлері

An infinitary logic is a logic that allows infinitely long statements and/or infinitely long 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 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.

As a language with infinitely long formulae is being presented, it is not possible to write such formulae down explicitly. To get around this problem a number of notational conveniences, which, strictly speaking, are not part of the formal language, are used. is used to point out an expression that is infinitely long. Where it is unclear, the length of the sequence is noted afterwards. Where this notation becomes ambiguous or confusing, suffixes such as are used to indicate an infinite disjunction over a set of formulae of cardinality The same notation may be applied to quantifiers, for example This is meant to represent an infinite sequence of quantifiers: a quantifier for each where
All usage of suffixes and are not part of formal infinitary languages. The axiom of choice is assumed (as is often done when discussing infinitary logic) as this is necessary to have sensible distributivity laws.

A theory is any set of sentences. The truth of statements in models are defined by recursion and will agree with the definition for finitary logic where both are defined. Given a theory T a sentence is said to be valid for the theory T if it is true in all models of T.

A logic in the language is complete if for every sentence S valid in every model there exists a proof of S. It is strongly complete if for any theory T for every sentence S valid in T there is a proof of S from T. An infinitary logic can be complete without being strongly complete. A cardinal is weakly compact when for every theory T in containing at most many formulas, if every S T of cardinality less than has a model, then T has a model. A cardinal is strongly compact when for every theory T in , without restriction on size, if every S T of cardinality less than has a model, then T has a model.

Two infinitary logics stand out in their completeness. These are the logics of and The former is standard finitary first order logic and the latter is an infinitary logic that only allows statements of countable size. The logic of is also strongly complete, compact and strongly compact. The logic of fails to be compact, but it is complete (under the axioms given above). Moreover, it satisfies a variant of the Craig interpolation property. If the logic of is strongly complete (under the axioms given above) then is strongly compact (because proofs in these logics cannot use or more of the given axioms).