Equational theory of positive numbers with exponentiation

Author:
R. Gurevič

Journal:
Proc. Amer. Math. Soc. **94** (1985), 135-141

MSC:
Primary 03C05; Secondary 03B25, 03C13, 03C65

MathSciNet review:
781071

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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 is decidable ( means for positive ) 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.

**[1]**Leon Henkin,*The logic of equality*, Amer. Math. Monthly**84**(1977), no. 8, 597–612. MR**0472649****[2]**A. G. Hovanskiĭ,*A class of systems of transcendental equations*, Dokl. Akad. Nauk SSSR**255**(1980), no. 4, 804–807 (Russian). MR**600749****[3]**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****[4]**D. Richardson,*Solution of the identity problem for integral exponential functions.*, Z. Math. Logik Grundlagen Math.**15**(1969), 333–340. MR**0262068****[5]**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

DOI:
https://doi.org/10.1090/S0002-9939-1985-0781071-1

Keywords:
Exponentiation of positive reals,
exponentiation of positive integers,
Tarski's high school algebra problem,
decidability of equational theory,
decidability of entailment relation,
differential ring,
finite model of Tarski's axioms

Article copyright:
© Copyright 1985
American Mathematical Society