Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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}$.


References [Enhancements On Off] (What's this?)

  • [1] H. Bachman, Transfinite Zahlen, 2nd ed., Springer-Verlag, Berlin, Heidelberg and New York, 1967. MR 0219424 (36:2506)
  • [2] N. G. de Bruijn, Asymptotic methods in analysis, North-Holland, Amsterdam, 1958.
  • [3] A. Ehrenfeucht, Polynomial functions with exponentiation are well ordered, Algebra Universalis 3 (1973), 261-262. MR 0332582 (48:10908)
  • [4] R. Gurevič, Transcendent numbers and eventual dominance of exponential functions, Abstracts Amer. Math. Soc. 4 (1983), 310.
  • [5] G. H. Hardy, Orders of infinity: The 'infinitärcalcül' of Paul du Bois-Reymond, 2nd ed., Cambridge Univ. Press, Cambridge, 1924.
  • [6] H. Levitz, An initial segment of the set of polynomial functions with exponentiation, Algebra Universalis 7 (1977), 133-136. MR 0498009 (58:16187)
  • [7] -, An ordinal bound for the set of polynomial functions with exponentiation, Algebra Universalis 8 (1978), 233-243. MR 473913 (81f:04003)
  • [8] -, The Cartesian product of sets and the Hessenberg natural product of ordinals, Czechoslovak Math. J. 29 (1974), 353-357. MR 536062 (81b:04003)
  • [9] -, Calculation of an order type: An application of non-standard methods, Z. Math. Logik Grundlag. Math. 28 (1982), 219-228. MR 668008 (83m:04004)
  • [10] D. Richardson, Solution of the identity problem for integral exponential functions, Z. Math. Logik Grundlag. Math. 15 (1969), 333-340. MR 0262068 (41:6678)
  • [11] T. Skolem, An ordered set of arithmetic functions representing the least $ \varepsilon $-number, Norske Vid. Selsk. Forh. (Trondheim) 29 (1956), 54-59. MR 0083957 (18:785a)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 03D20, 06F05, 26A12

Retrieve articles in all journals with MSC: 03D20, 06F05, 26A12


Additional Information

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

American Mathematical Society