Equational theory of positive numbers with exponentiation

Gurevič, R.

doi:10.1090/S0002-9939-1985-0781071-1

Equational theory of positive numbers with exponentiation
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by R. Gurevič

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

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

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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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