The Real Numbers as a Wreath Product

Abstract

This chapter discusses the real numbers as a wreath product. Few mathematical structures have undergone as many revisions or have been presented in as many guises as the real numbers. Every generation re-examines the real numbers in the light of its values and mathematical objectives. It is often deplored that the field of real numbers is not constructive in any of the currently accepted meanings of the word. A real number turns out to be an equivalence class of such formal Laurent series, or strings as one call them, and arithmetic operations are performed on these equivalence classes. In this way, addition and multiplication become carryless. The chapter discusses the construction of real numbers. Real numbers can be constructed by algorithmically describing the operations of binary addition, multiplication, and division on infinite strings of zeroes and ones. It turns out, however, that an explicit description of the algorithms for the elementary operations is very cumbersome, owing to the presence of carries. The digits of a string used to represent a real number are allowed to be arbitrary integers. Two strings that differ only in that a carry has been performed on one of them are decreed equivalent, and a real number is an equivalence class under iterated performances of carry operations.