Surreal Numbers
In mathematics , the surreal number system is a totally ordered class containing the real numbers as well as infinite and infinitesimal numbers , respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field . [a] If formulated in Von Neumann–Bernays–Gödel set theory , the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the rational functions , the Levi-Civita field , the superreal numbers , and the hyperreal numbers , can be realized as subfields of the surreals. [1] It has also been shown (in Von Neumann–Bernays–Gödel set theory ) that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the axiom of global choice , this need not be the case, and in such the...