Introduction to infinitesimals

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Keywords: EndGame

Introduction

In Combinatorial Game Theory (CGT) an infinitesimal is a non-zero combinatorial game in which both the Left stop and the Right stop are 0. In CGT you win if your opponent has no move, so play with infinitesimals is concerned with getting the last move (tedomari). A great deal of CGT research has been concerned with play at temperature 0.

The simplest infinitesimal is ast (star), also written

 { 0 | 0 }

The reason they are called infinitesimals is that even if such positions are advantageous to you (e.g., you win no matter who moves first), no finite number of such positions added together will be able to beat your opponent if he has a single point, or even a fraction of a point. In other words, you can have games which are positive but less than any positive rational number, which can be taken as a definition of the mathematical concept of "infinitesimal." Other infinitesimals, like ast, are confused with zero. That is, they are smaller than any positive game and larger than any negative game, but are not larger than zero. Games that are smaller than any positive number and larger than any negative number are known as small games.

For the application of infinitesimals to go, see Go Infinitesimals and Chilling.