Infinitary logic

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An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs.[1] The concept was introduced by Zermelo in the 1930s.[2]

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.[3]

A word on notation and the axiom of choice

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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.

Formal languages

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A first-order infinitary language  ,   regular,   or  , has the same set of symbols as a finitary logic and may use all the rules for formation of formulae of a finitary logic together with some additional ones:[4]

  • Given a set of formulae   with   then   and   are formulae. (In each case the sequence has length  .)
  • Given a set of variables   with   and a formula   then   and   are formulae. (In each case the sequence of quantifiers has length  .)

The language may also have function, relation, and predicate symbols of finite arity.[5] Karp also defined languages   with   an infinite cardinal and some more complicated restrictions on   that allow for function and predicate symbols of infinite arity, with   controlling the maximum arity of a function symbol and   controlling predicate symbols.[6]

The concepts of free and bound variables apply in the same manner to infinite formulae. Just as in finitary logic, a formula all of whose variables are bound is referred to as a sentence.

Definition of Hilbert-type infinitary logics

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A theory   in infinitary language   is a set of sentences in the logic. A proof in infinitary logic from a theory   is a (possibly infinite) sequence of statements that obeys the following conditions: Each statement is either a logical axiom, an element of  , or is deduced from previous statements using a rule of inference. As before, all rules of inference in finitary logic can be used, together with an additional one:

  • Given a set of statements   that have occurred previously in the proof then the statement   can be inferred.[7]

If  , forming universal closures may not always be possible, however extra constant symbols may be added for each variable with the resulting satisfiability relation remaining the same.[8] To avoid this, some authors use a different definition of the language   forbidding formulas from having more than   free variables.[9]

The logical axiom schemata specific to infinitary logic are presented below. Global schemata variables:   and   such that  .

  •  
  • For each  ,  
  • Chang's distributivity laws (for each  ):  , where   or  , and  
  • For  ,  , where   is a well ordering of  

The last two axiom schemata require the axiom of choice because certain sets must be well orderable. The last axiom schema is strictly speaking unnecessary, as Chang's distributivity laws imply it,[10] however it is included as a natural way to allow natural weakenings to the logic.

Completeness, compactness, and strong completeness

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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.

Concepts expressible in infinitary logic

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In the language of set theory the following statement expresses foundation:

 

Unlike the axiom of foundation, this statement admits no non-standard interpretations. The concept of well-foundedness can only be expressed in a logic that allows infinitely many quantifiers in an individual statement. As a consequence many theories, including Peano arithmetic, which cannot be properly axiomatised in finitary logic, can be in a suitable infinitary logic. Other examples include the theories of non-archimedean fields and torsion-free groups.[11][better source needed] These three theories can be defined without the use of infinite quantification; only infinite junctions[12] are needed.

Truth predicates for countable languages are definable in  .[13]

Complete infinitary logics

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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).

References

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  1. ^ Moore, Gregory H. (1997). "The prehistory of infinitary logic: 1885–1955". In Dalla Chiara, Maria Luisa; Doets, Kees; Mundici, Daniele; van Benthem, Johan (eds.). Structures and Norms in Science. Springer-Science+Business Media. pp. 105–123. doi:10.1007/978-94-017-0538-7_7. ISBN 978-94-017-0538-7.
  2. ^ Kanamori, Akihiro (2004). "Zermelo and set theory" (PDF). The Bulletin of Symbolic Logic. 10 (4): 487–553. doi:10.2178/bsl/1102083759. Retrieved 22 August 2023.
  3. ^ Woodin, W. Hugh (2011). "The Continuum Hypothesis, the generic-multiverse of sets, and the Ω Conjecture". In Kennedy, Juliette; Kossak, Roman (eds.). Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies. Cambridge University Press. pp. 13–42. doi:10.1017/CBO9780511910616.003. ISBN 978-0-511-91061-6. Archived from the original on 1 March 2024. Retrieved 1 March 2024.
  4. ^ Karp 1964, pp. 1–2.
  5. ^ Karp 1964, p. 1.
  6. ^ Karp 1964, pp. 101–102.
  7. ^ Karp 1964, pp. 39–54.
  8. ^ Karp 1964, p. 127.
  9. ^ J. L. Bell, "Infinitary Logic". Stanford Encyclopedia of Philosophy, revised 2023. Accessed 26 July 2024.
  10. ^ Chang, C. C. (1957). "On the representation of α-complete Boolean algebras". Transactions of the American Mathematical Society. 85 (1): 208–218. doi:10.1090/S0002-9947-1957-0086792-1.
  11. ^ Rosinger, Elemer E. (2010). "Four departures in Mathematics and Physics". arXiv:1003.0360 [math.GM]. {{cite arXiv}}: Unknown parameter |citeseerx= ignored (help)
  12. ^ Bennett, David W. (1980). "Junctions". Notre Dame Journal of Formal Logic. 21 (1): 111–118. doi:10.1305/ndjfl/1093882943.
  13. ^ Pogonowski, Jerzy (10 June 2010). "Inexpressible longing for the intended model" (PDF). Zakład Logiki Stosowanej. Uniwersytet im. Adama Mickiewicza w Poznaniu. p. 4. Retrieved 1 March 2024.

Sources

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