Equational theory of positive numbers with exponentiation

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 (N, 1, +, , ) is decidable (a f b means ah 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.