Fine structure of the integral exponential functions below 2^{2^{𝑥}}

Dahn, Bernd I.

doi:10.1090/S0002-9947-1986-0854094-7

Fine structure of the integral exponential functions below $2^ {2^ x}$
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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.

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