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String theory

Published online by Cambridge University Press:  12 March 2014

John Corcoran,
William Frank and
Michael Maloney
John Corcoran
Affiliation:
State University of New York at Buffalo, Amherst, New York 14226
William Frank
Affiliation:
State University of New York at Buffalo, Amherst, New York 14226
Michael Maloney
Affiliation:
State University of New York at Buffalo, Amherst, New York 14226
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Abstract

For each n > 0, two alternative axiomatizations of the theory of strings over n alphabetic characters are presented. One class of axiomatizations derives from Tarski's system of the Wahrheitsbegriff and uses the n characters and concatenation as primitives. The other class involves using n character-prefixing operators as primitives and derives from Hermes' Semiotik. All underlying logics are second order. It is shown that, for each n, the two theories are synonymous in the sense of deBouvere. It is further shown that each member of one class is synonymous with each member of the other class; thus that all of the theories are synonymous with each other and with Peano arithmetic. Categoricity of Peano arithmetic then implies categoricity of each of the above theories.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 39 , Issue 4 , December 1974, pp. 625 - 637
Copyright
Copyright © Association for Symbolic Logic 1974

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References

REFERENCES

[1] Carnap, Rudolf, Logical syntax of language, 1934; translation by Amethe Smeaton, New York, 1959.Google Scholar
[2] Chomsky, Noam, Syntactic structures, The Hague, 1957.CrossRefGoogle Scholar
[3] Church, Alonzo, Introduction to mathematical logic (based on notes by C. A. Truesdell), Princeton, 1944.Google Scholar
[4] Church, Alonzo, Introduction to mathematical logic, Princeton, 1956.Google Scholar
[5] Corcoran, John, A semantic definition of definition, presentation to Chicago meeting of Association for Symbolic Logic (1967) abstracted in this Journal, vol. 36 (1971), pp. 366–367.Google Scholar
[6] Corcoran, John, Conceptual structure of classical logic, Philosophy and Phenomenological Research, vol. 33 (1972), pp. 25–47.Google Scholar
[7] deBouvere, K. L., Logical synonymity, Indagationes Mathematica, vol. 27 (1965).Google Scholar
[8] deBouvere, K. L., Synonymous theories, The theory of models (Proceedings of the 1963 International Symposium at Berkeley), North-Holland, Amsterdam, 1965.Google Scholar
[9] Dedekind, Richard, The nature and meaning of numbers (written before 1893), Essays on the theory of numbers; translation by W. W. Beman, New York, 1963.Google Scholar
[10] Henkin, Leon, Banishing the rule of substitution for functional variables, this Journal, vol. 18 (1953), pp. 201–208.Google Scholar
[11] Hermes, Hans, Semiotik, Eine Theorie der Zeichengestalten als Grundlage für Untersuchungen von formalizierten Sprachen, Forschungen zur Logik und zur Grundlage der exakten Wissenschaften, n.s. no. 5, Leipzig, 1938.Google Scholar
[12] Hilbert, David, The foundations of mathematics, 1927; reprinted in van Heijenoort, From Frege to Gödel, Cambridge, Mass., 1967.Google Scholar
[13] Kleene, Stephen Cole, Introduction to metamathematics, New York, 1952.Google Scholar
[14] Maloney, Michael J., Logical and axiomatic foundations for the study of formal languages and symbolic computation, unpublished Ph.D. dissertation, University of Pennsylvania, 1969. Available from University Microfilms, Ann Arbor, Michigan.Google Scholar
[15] Martin, Richard, Truth and denotation, London, 1958.Google Scholar
[16] Montague, Richard, Set theory and higher order logic in Formal systems and recursive functions (Crossley, and Dummett, , Editors), North-Holland, Amsterdam, 1965.Google Scholar
[17] Quine, Willard, Concatenation as a basis for arithmetic, this Journal, vol. 11 (1946).Google Scholar
[18] Quine, Willard, Mathematical logic, revised edition, Cambridge, Mass., 1951.CrossRefGoogle Scholar
[19] Robbin, Joel, Mathematical logic, New York, 1969.Google Scholar
[20] Tarski, Alfred, Undecidable theories (in collaboration with A. Mostowski and R. Robinson), North-Holland, Amsterdam, 1953.Google Scholar
[21] Tarski, Alfred, On definable sets of real numbers (1931), Logic, semantics and metamathematics, Oxford, 1956.Google Scholar
[22] Tarski, Alfred, The concept of truth in formalized languages (1934), Logic, semantics and metamathematics, Oxford, 1956.Google Scholar