Abstract
The authors analyze Diophantine sets and show that all recursively enumerable sets are Diophantine. Based on the classical results from the theory of recursive functions, a simple version of the theorem on the incompleteness of arithmetic is provided: there is a polynomial that has no positive integer solutions, and for which it is impossible to prove the absence of positive roots.
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References
Yu. V. Matiyasevich, “Diophantine sets,” Uspekhi Mat. Nauk, Vol. 2, Iss. 5, 185–222 (1971).
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Yu. V. Matiyasevich, Hilbert’s Tenth Problem [in Russian], Nauka, Moscow (1993).
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Translated from Kibernetyka ta Systemnyi Analiz, No. 5, September–October, 2023, pp. 16–21.
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Gupal, A.M., Vagis, O.A. Incompleteness of Arithmetic from the Viewpoint of Diophantine Set Theory. Cybern Syst Anal 59, 698–703 (2023). https://doi.org/10.1007/s10559-023-00605-y
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DOI: https://doi.org/10.1007/s10559-023-00605-y