Fine structure of the integral exponential functions below $2^ {2^ x}$
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- by Bernd I. Dahn PDF
- Trans. Amer. Math. Soc. 297 (1986), 707-716 Request permission
Abstract:
Integral exponential functions are the members of the least class of real functions containing $1$, the identity function, and closed under addition, multiplication, and binary exponentiation sending $f$ and $g$ to ${f^g}$. This class is known to be wellordered by the relation of eventual dominance. It is shown that for each natural number $n$ the order type of the integral exponential functions below ${2^{{x^n}}}$ (below ${x^{{x^n}}}$) is exactly ${\omega ^{{\omega ^{2n - 1}}}}$ (${\omega ^{{\omega ^{2n}}}}$ respectively). The proof, using iterated asymptotic expansions, contains also a new proof that integral exponential functions below ${2^{{2^x}}}$ are wellordered.References
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Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 297 (1986), 707-716
- MSC: Primary 26A12; Secondary 03D20, 04A99
- DOI: https://doi.org/10.1090/S0002-9947-1986-0854094-7
- MathSciNet review: 854094