Stanford Encyclopedia of Philosophy (2008)
AbstractTraditionally, 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
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Citations of this work
SAD Computers and Two Versions of the Church–Turing Thesis.Tim Button - 2009 - British Journal for the Philosophy of Science 60 (4):765-792.
Alfred Tarski and the "Concept of Truth in Formalized Languages": A Running Commentary with Consideration of the Polish Original and the German Translation.Monika Gruber - 2016 - Cham, Switzerland: Springer Verlag.
An Argument for the Ontological Innocence of Mereology.Rohan French - 2016 - Erkenntnis 81 (4):683-704.
Grounding, Quantifiers, and Paradoxes.Francesco A. Genco, Francesca Poggiolesi & Lorenzo Rossi - 2021 - Journal of Philosophical Logic 50 (6):1417-1448.
“Mathematics is the Logic of the Infinite”: Zermelo’s Project of Infinitary Logic.Jerzy Pogonowski - 2021 - Studies in Logic, Grammar and Rhetoric 66 (3):673-708.
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