Stanford Encyclopedia of Philosophy (2008)

Authors
John L. Bell
University of Western Ontario
Abstract
Traditionally, expressions in formal systems have been regarded as signifying finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via "Gödel numbering") and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning "languages" some of whose formulas would be naturally identified as infinite sets . A "language" of this kind is called an infinitary language : in this article I discuss those infinitary languages which can be obtained in a straightforward manner from first-order languages by allowing conjunctions, disjunctions and, possibly, quantifier sequences, to be of infinite length. In the course of the discussion it will be seen that, while the expressive power of such languages far exceeds that of their finitary (first-order) counterparts, very few of them possess the "attractive" features (e.g., compactness and completeness) of the latter. Accordingly, the infinitary languages that do in fact possess these features merit special attention.
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References found in this work BETA

Admissible Sets and Structures.Jon Barwise - 1978 - Studia Logica 37 (3):297-299.
Set Theory: An Introduction to Large Cardinals.F. R. Drake & T. J. Jech - 1976 - British Journal for the Philosophy of Science 27 (2):187-191.
[Omnibus Review].H. Jerome Keisler - 1970 - Journal of Symbolic Logic 35 (2):342-344.
Logic with Denumerably Long Formulas and Finite Strings of Quantifiers.Dana Scott - 1965 - In J. W. Addison (ed.), Journal of Symbolic Logic. Amsterdam: North-Holland Pub. Co.. pp. 1104--329.

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SAD Computers and Two Versions of the Church–Turing Thesis.Tim Button - 2009 - British Journal for the Philosophy of Science 60 (4):765-792.

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