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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


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 $ ({\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.

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Additional Information

PII: S 0002-9939(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

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