Celebratio Mathematica — van den Dries

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Works connected to Hilbert Levitz

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L. van den Dries and H. Levitz: “On Skolem’s exponential functions below \( 2^{2^X} \),” Trans. Am. Math. Soc. 286 : 1 (November 1984), pp. 339–349. MR 756043 Zbl 0556.03036 article

A result of Ehrenfeucht implies that the smallest class of number-theoretic functions \[ f:\mathbf{N}\to\mathbf{N} \] containing the constants \( 0,\,1,\,2,\dots \), the identity function \( X \), and closed under addition, multiplication and \( f\to f^X \), is well-ordered by the relation of eventual dominance. We show that its order type is \( \omega^{\omega^{\omega}} \), and that for any two nonzero functions \( f,\,g \) in the class the quotient \( f(n)/g(n) \) tends to a limit in \[ E^+\cup\{0,\infty\} \] as \( n\to\infty \), where \( E^+ \) is the smallest set of positive real numbers containing 1 and closed under addition, multiplication and under the operations \( x\to x^{-1} \), \( x\to e^x \).