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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Equational theory of positive numbers with exponentiation
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by R. Gurevič PDF
Proc. Amer. Math. Soc. 94 (1985), 135-141 Request permission

Abstract:

A. Tarski asked if all true identities involving 1, addition, multiplication, and exponentiation can be derived from certain so-called "high-school" identities (and a number of related questions). I prove that equational theory of $({\mathbf {N}},1, + , \cdot , \uparrow )$ is decidable ($a \uparrow b$ means ${a^b}$ for positive $a,b$) and that entailment relation in this theory is decidable (and present a similar result for inequalities). A. J. Wilkie found an identity not derivable from Tarski’s axioms with a difficult proof-theoretic argument of nonderivability. I present a model of Tarski’s axioms consisting of 59 elements in which Wilkie’s identity fails.
References
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  • Angus Macintyre, The laws of exponentiation, Model theory and arithmetic (Paris, 1979–1980) Lecture Notes in Math., vol. 890, Springer, Berlin-New York, 1981, pp. 185–197. MR 645003
  • D. Richardson, Solution of the identity problem for integral exponential functions, Z. Math. Logik Grundlagen Math. 15 (1969), 333–340. MR 262068, DOI 10.1002/malq.19690152003
  • Alex J. Wilkie, On exponentiation—a solution to Tarski’s high school algebra problem, Connections between model theory and algebraic and analytic geometry, Quad. Mat., vol. 6, Dept. Math., Seconda Univ. Napoli, Caserta, 2000, pp. 107–129. MR 1930684
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 94 (1985), 135-141
  • MSC: Primary 03C05; Secondary 03B25, 03C13, 03C65
  • DOI: https://doi.org/10.1090/S0002-9939-1985-0781071-1
  • MathSciNet review: 781071