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Three universal representations of recursively enumerable sets

Published online by Cambridge University Press:  12 March 2014

James P. Jones
James P. Jones*
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary T2N 1N4, Canada
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Extract

In his celebrated paper of 1931 [7], Kurt Gödel proved the existence of sentences undecidable in the axiomatized theory of numbers. Gödel's proof is constructive and such a sentence may actually be written out. Of course, if we follow Gödel's original procedure the formula will be of enormous length.

Forty-five years have passed since the appearance of Gödel's pioneering work. During this time enormous progress has been made in mathematical logic and recursive function theory. Many different mathematical problems have been proved recursively unsolvable. Theoretically each such result is capable of producing an explicit undecidable number theoretic predicate. We have only to carry out a suitable arithmetization. Until recently, however, techniques were not available for carrying out these arithmetizations with sufficient efficiency.

In this article we construct an explicit undecidable arithmetical formula, F(x, n), in prenex normal form. The formula is explicit in the sense that it is written out in its entirety with no abbreviations. The formula is undecidable in the recursive sense that there exists no algorithm to decide, for given values of n, whether or not F(n, n) is true or false. Moreover F(n, n) is undecidable in the formal (axiomatic) sense of Gödel [7]. Given any of the usual axiomatic theories to which Gödel's Incompleteness Theorem applies, there exists a value of n such that F(n, n) is unprovable and irrefutable. Thus Gödel's Incompleteness Theorem can be “focused” into the formula F(n, n). Thus some substitution instance of F(n, n) is undecidable in Peano arithmetic, ZF set theory, etc.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 2 , June 1978 , pp. 335 - 351
Copyright
Copyright © Association for Symbolic Logic 1978

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References

REFERENCES

[1]Čudnovskiï, G. V., Diophantine predicates (Russian), Uspehi Matematičeskih Nauk, vol. 25 (1970), pp. 185–186.Google Scholar
[2]Davis, Martin, Hubert's tenth problem is unsolvable, American Mathematical Monthly, vol. 80 (1973), pp. 233–269.CrossRefGoogle Scholar
[3]Davis, Martin, Computability and unsolvabittty, McGraw-Hill, N. Y., 1958.Google Scholar
[4]Davis, Martin, An explicit Diophantine definition of the exponential function, Communications on Pure and Applied Mathematics, vol. 24 (1971), pp. 137–145.CrossRefGoogle Scholar
[5]Davis, Martin, Putnam, Hilary and Robinson, Julia, The decision problem for exponential Diophantine equations, Annals of Mathematics, vol. 74 (1961), pp. 425–436 = Matematika 8:5 (1964), pp. 69–79. MR24#A3061.CrossRefGoogle Scholar
[6]Davis, Martin, Matijasevič, Yuri and Robinson, Julia, Hubert's tenth problem. Diophantine equations: positive aspects of a negative solution, Proceedings of the Conference on the Hilbert Problems, Dekalb, Illinois, May 1974, American Mathematical Society, Providence, RI, 1976, pp. 323–378.Google Scholar
[7]Godel, Kurt, Über formal Unentscheidbare Sätze der Principia Mathematica und Verwandter Systeme I, Monatshefte Mathematik und Physik, vol. 38 (1931), pp. 173–198. English translations: (1) On formally undecidable propositions of Principia Mathematica and related systems (Martin Davis, Editor), The undecidable, Raven Press, 1965, pp. 5–38; (2) From Frege to Gödel (Jean Van Heijenoort, Editor), Harvard University Press, 1967, pp. 596–616.CrossRefGoogle Scholar
[8]Hilbert, David, Mathematische Probleme, Vortrag, gehalten auf dem intemationalen Mathematiker-Kon-gress zu Paris 1900, Nachrichten der Akademie der Wissenschaften in Gbttingen. II. Mathematische-Physikalische Klasse, 1900, pp. 253–297. English translation: Bulletin of the American Mathematical Society, vol. 8 (1901–1902), pp. 437–479.Google Scholar
[9]Jones, J. P., Diophantine representation of the Fibonacci numbers, Fibonacci Quarterly, vol. 13 (1975), pp. 84–88. MR 52 #3035.Google Scholar
[10]Jones, J. P., Diophantine representation of the Lucas numbers, Fibonacci Quarterly, vol. 14 (1976), p. 134.Google Scholar
[11]Jones, J.P., Sato, D., Wada, H. and Wiens, D.P., Diophantine representation of the set ofprime numbers, American Mathematical Monthly, vol. 83 (1976), pp. 449–464.CrossRefGoogle Scholar
[12]Jones, J. P., Diophantine representation of Mersenne and Fermat primes, Acta Arithmetica (in press)Google Scholar
[13]Kosovskiĭ, N. K., On Diophantine representation of the sequence of solutions of Pell's equation (Russian), Zapiski Naučnyh Seminarov Leningradskogo Otdelenija Matematičeskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, vol. 20 (1971), pp. 49–59. English translation: Journal of Soviet Mathematics, vol. 1 (1973), pp. 28–35.Google Scholar
[14]Matijasevič, Yu. V., Enumerable sets are Diophantine (Russian), Doklady Akademii Nauk SSSR, vol. 191 (1970), pp. 279–282. English translation: Soviet Mathematics, Doklady, vol. 11 (1970), pp. 354–358.Google Scholar
[15]Matijasevič, Yu. V., Diophantine representation of enumerable predicates (Russian), lzvestija Akademii Nauk SSSR, Serija Matematika, vol. 35 (1971), pp. 3–30. English translation: Mathematics of the USSR–Izvestija, vol. 5 (1971), pp. 1–28.Google Scholar
[16]Matijasevič, Yu. V., Diophantine sets (Russian), Uspehi Matematičeskih Nauk, vol. 27 (1972), pp. 185–222. English translation: Russian Mathematical Surveys, vol. 27 (1972), pp. 124–164.Google Scholar
[17]Matijasevič, Yu. V., Arithmetical representation of enumerable sets with a small number of quantifiers (Russian), Zapiski Naučnyh Seminarov Leningradskogo Otdelenija Matematičeskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, vol. 32 (1972), pp. 77–84.Google Scholar
[18]Matijasevič, Yu. V. and Robinson, Julia, Two universal 3-quantifier representations of r.e. sets (Russian), Theory of algorithms and mathematical logic. Computation Centre, Akademii Nauk SSSR, Moscow, 1974, pp. 112–123 (Volume dedicated to A. A. Markov).Google Scholar
[19]Matijasevič, Yu. V., On recursive unsolvability of Hubert's tenth problem, Proceedings of the Fourth International Congress on Logic, Methodology and Philosophy of Science, Bucharest, 1971, North-Holland, Amsterdam, 1973, pp. 89–110.CrossRefGoogle Scholar
[20]Matijasevič, Yu. V. and Robinson, Julia, Reduction of an arbitrary Diophantine equation to one in 13 unknowns, Acta Arithmetica, vol. 27 (1975), pp. 521–553.CrossRefGoogle Scholar
[21]Putnam, Hilary, An unsolvable problem in number theory, this Journal, vol. 25 (1960), pp. 220–232 = Matematika, vol. 8 (1964), pp. 55–67.Google Scholar
[22]Robinson, Julia, Existential definability in arithmetic, Transactions of the American Mathematical Society, vol. 72 (1952), pp. 437–449 = Matematika, vol. 8 (1964), pp. 3–14. MR 14, 4.CrossRefGoogle Scholar
[23]Robinson, Julia, Hubert's tenth problem, Proceedings of the Symposium on Pure Mathematics, vol. 20 (1971), pp. 191–194.CrossRefGoogle Scholar
[24]Robinson, Raphael M., Arithmetical representation of recursively enumerable sets, this Journal, vol. 21 (1956), pp. 162–186.Google Scholar
[25]Robinson, Raphael M., Some representations of Diophantine sets, this Journal, vol. 37 (1972), pp. 572–578.Google Scholar
[26]Siegel, Carl L., Zur theorie der quadratischen formen, Nachrichten der Akademie Wissenschaften in Göttingen. II. Mathematische-Physikalische Klasse, 1972, pp. 21–46.Google Scholar
[27]Skolem, Thorale, Diophantische Gleichungen, Springer, Berlin, 1938.Google Scholar