Fine structure of the integral exponential functions below $2\sp {2\sp x}$

Bernd I. Dahn
1986 Transactions of the American Mathematical Society  
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 / and g to fg. 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 2X (below xx ) is exactly uu (ui" respectively). The proof, using iterated asymptotic expansions, contains
more » ... lso a new proof that integral exponential functions below 22 are wellordered. In 1956 Skolem [Sk] investigated the class of integral exponential functions-the least class containing 1, the identity function, and closed under additon, multiplication and binary exponentiation sending /, g to f9. This class of functions, denoted by Sk, is linearly ordered by the relation < of eventual dominance (see [H]). Skolem asked whether this is a wellordering and-if so-what is its order type, he pointed out a subset of Sk having the order type £o = sup{ui,^,^ ,■■■}■ Ehrenfeucht [E] proved in 1973 that the eventual dominance relation is in fact a wellordering. He applied a powerful graph theoretical result due to Kruskal. Recently van den Dries and Levitz [DL] showed that the integral exponential functions below 22X are wellordered of type u" . Putting for / G Sk Sk(/) := {g G Sk: g < /}, their proof gives also that 2n"<|Sk(2^)|<^n2 for each nGw, It is the aim of this paper to show that in fact c^2"" = |Sk(2*")|. It is worth noting that we shall not refer to Ehrenfeucht's paper [E] anymore, i.e., we will also obtain a new'proof that Sk(221) is wellordered. The methods of the present paper were inspired by [DL] and came up in numerous discussions with Peter Goring, whom the author thanks very much for his patience and active interest. For the reader it will be helpful to recall from [D] and [DG] that exponentiallogarithmic terms, especially integral exponential functions, can be represented in a canonic way by iterated powerseries of length co or, optionally, as powerseries of transfinite length. Let us first state some conventions and notations. If K, L are ordered fields, K Ç L, a,b G L, then we write a«i mod K iff b ^ 0 and \a/b\ < e for each
doi:10.1090/s0002-9947-1986-0854094-7 fatcat:nrhgfxgn3fa4xbgosegnnrjpdu