Foundations of Analysis over Surreal Number Fields

Foundations of Analysis over Surreal Number Fields

1st Edition - April 1, 1987

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  • Author: N.L. Alling
  • eBook ISBN: 9780080872520

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In this volume, a tower of surreal number fields is defined, each being a real-closed field having a canonical formal power series structure and many other higher order properties. Formal versions of such theorems as the Implicit Function Theorem hold over such fields. The Main Theorem states that every formal power series in a finite number of variables over a surreal field has a positive radius of hyper-convergence within which it may be evaluated. Analytic functions of several surreal and surcomplex variables can then be defined and studied. Some first results in the one variable case are derived. A primer on Conway's field of surreal numbers is also given.Throughout the manuscript, great efforts have been made to make the volume fairly self-contained. Much exposition is given. Many references are cited. While experts may want to turn quickly to new results, students should be able to find the explanation of many elementary points of interest. On the other hand, many new results are given, and much mathematics is brought to bear on the problems at hand.

Table of Contents

  • Introduction. 1. Preliminaries. 2. The &xgr;-Topology. 3. The &xgr;-Topology on Affine n-Space. 4. Introduction to the Surreal Field No. 5. The Surreal Fields &xgr;No, and Related Topics. 6. The Valuation Theory of Ordered Fields, Applied to No and &xgr;No. 7. Power Series: Formal and Hyper-Convergent. 8. A Primer on Analytic Functions of a Surreal Variable. Bibliography. Index.

Product details

  • No. of pages: 372
  • Language: English
  • Copyright: © North Holland 1987
  • Published: April 1, 1987
  • Imprint: North Holland
  • eBook ISBN: 9780080872520

About the Author

N.L. Alling

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  • Alec R. Wed Feb 21 2018

    Beautiful beginnings of a theory

    Alling's book provides an excellent first pass at a theory of analysis over the Surreal numbers, utilizing tools from classical ordered group/ring valuation theory together with set theory. Although the exponential function developed within only holds for infinitesimal Surreals, and a more comprehensive exponential theory is developed in other texts on Surreals, Alling's construction of Surreal numbers as Hahn series over the reals is fascinating and enlightening -- a must read for anyone wanting to gain a foothold in the strange world of Surreal analysis. There are minor typos in the text, some of them significant in proofs, for which I have deducted one star.