Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
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.


References [Enhancements On Off] (What's this?)


Similar Articles

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