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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On Skolem's exponential functions below $ 2\sp{2\sp{X}}$


Authors: Lou van den Dries and Hilbert Levitz
Journal: Trans. Amer. Math. Soc. 286 (1984), 339-349
MSC: Primary 03D20; Secondary 06F05, 26A12
DOI: https://doi.org/10.1090/S0002-9947-1984-0756043-7
MathSciNet review: 756043
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Abstract: 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, \ldots $, 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}$.


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DOI: https://doi.org/10.1090/S0002-9947-1984-0756043-7
Article copyright: © Copyright 1984 American Mathematical Society