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

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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.

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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