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Concatenation as a basis for arithmetic

Published online by Cambridge University Press:  12 March 2014

W. V. Quine*
Affiliation:
Harvard University

Extract

General syntax, the formal part of the general theory of signs, has as its basic operation the operation of concatenation, expressed by the connective ‘⌢’ and understood as follows : where x and y are any expressions, xy is the expression formed by writing the expression x immediately followed by the expression y. E.g., where ‘alpha’ and ‘beta’ are understood as names of the respective signs ‘α’ and ‘β’, the syntactical expression ‘alpha⌢beta’ is a name of the expression ‘αβ’.

Tarski and Hermes have presented axioms for concatenation, and definitions of derivative syntactical concepts. Hermes has also related concatenation theory to the arithmetic of natural numbers, constructing a model of the latter within the former. Conversely, Gödel's proof of the impossibility of a complete consistent systematization of arithmetic depended on constructing a model of concatenation theory within arithmetic.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1947

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References

1 Tarski, Alfred, Der Wahrheitsbegriff in den formalisierten Sprachen, Studia philosophica (Lwów), vol. 1 (1935), pp. 261405 Google Scholar, esp. pp. 287 ff.

2 Hermes, Hans, Semiotik. Eine Theorie der Zeichengestalten als Grundlage für Untersuchungen von formalisierten Sprachen. Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, new series, no. 5. Leipzig, 1938, 22 pp.Google Scholar

3 Gödel, Kurt, Über formal unentschuldbare Sätze der Principia Mathematica und verwandter Systeme, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173198.CrossRefGoogle Scholar

4 Quine, W. V., Definition of substitution, Bulletin of the American Mathematical Society, vol. 42 (1936), pp. 561569 CrossRefGoogle Scholar; On derivability, this Journal, vol. 2 (1937), pp. 113-119; Mathematical logic (New York, 1940), Ch. 7.

5 The definiens in D1 could be simplified to if we were to assume a null sequence among the values of our variables. However, I have chosen rather to repudiate the null sequence throughout the present paper, lest it be thought to be an essential assumption.

6 The reasoning behind it is evident from On derivabilily, this Journal, vol. 2 (1937), p. 115, where the present identification of integers with sequences is so rephrased as to exhibit its relation to the dual system of numeration.

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