Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Equational theory of positive numbers with exponentiation
HTML articles powered by AMS MathViewer

by R. Gurevič PDF
Proc. Amer. Math. Soc. 94 (1985), 135-141 Request permission


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.
  • Leon Henkin, The logic of equality, Amer. Math. Monthly 84 (1977), no. 8, 597–612. MR 472649, DOI 10.2307/2321009
  • A. G. Hovanskiĭ, A class of systems of transcendental equations, Dokl. Akad. Nauk SSSR 255 (1980), no. 4, 804–807 (Russian). MR 600749
  • 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
Similar Articles
Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 94 (1985), 135-141
  • MSC: Primary 03C05; Secondary 03B25, 03C13, 03C65
  • DOI:
  • MathSciNet review: 781071