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

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

MathSciNet review:
781071

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

- Leon Henkin,
*The logic of equality*, Amer. Math. Monthly**84**(1977), no. 8, 597–612. MR**472649**, DOI https://doi.org/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 https://doi.org/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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
03C05,
03B25,
03C13,
03C65

Retrieve articles in all journals with MSC: 03C05, 03B25, 03C13, 03C65

Additional Information

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