num_pages: 58
description: Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 57. Chapters: Countable set Cantor's diagonal argument Surreal number Continuum hypothesis Hyperreal number Extended real number line Uncountable set Where Mathematics Comes From Absolute Infinite Ultrafinitism Infinitesimal Infinite monkey theorem Actual infinity Non-standard calculus Real projective line Temporal finitism Cardinality of the continuum Aleph number Beth number Line at infinity Plane at infinity Apeirogon Levi-Civita field Point at infinity Hyperplane at infinity Infinity plus one Superreal number Hyperinteger Circular points at infinity Directed infinity Amorphous set. Excerpt: In mathematics the surreal number system is an arithmetic continuum 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 a total order and the usual arithmetic operations (addition subtraction multiplication and division); as such they form an ordered field. In a rigorous set theoretic sense 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 are subfields of the surreals. The surreals also contain all transfinite ordinal numbers reachable in the set theory in which they are constructed. The definition and construction of the surreals is due to John Horton Conway. They were introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. This book is a mathematical novelette and is notable as one of the rare cases where a new mathematical idea was first presented in a work of fiction. In his book which takes the...
language: English
title: Infinity: Countable Set, Cantor's Diagonal Argument, Surreal Number, Continuum Hypothesis, Hyperreal Number, Extended Real Number Line
publisher: Books LLC, Wiki Series