A structure is called a collection of games if and where is the power set of and . The elements of are called games and the convention here is that they would be denoted by the upper case Latin letters G,H,K,... .

Define the binary relation, R (for reachable) between and itself by iff .

is called loopy if where is the transitive closure of R. Otherwise, it's called nonloopy.If there exists an element 0 of , with , then we call it the zero element.

*Lemma 1*: The zero element, if it exists is unique.

Table of contents |

2 Nimbers 3 Surreal numbers |

*Lemma 2*: If is finite and nonloopy, then it contains a zero element.

Let be the smallest collection of games containing 0 and .

*Lemma 3*: All finite nonloopy games are isomorphic to a subcollection of .

So, without any loss of generality, we can work solely with .

We can define a binary operator recursively by

and .

*Lemma 5*: .

The set of second-player-win games, P is defined recursively as follows:
*I'll add the definition later; it's getting late*

The negative of a game is defined recursively as follows: .

*Lemma 6*: This definition is well-defined and unique.

The relation is defined by iff .

*Lemma 7*: is an equivalence relation.

*Lemma 8*: .

*Lemma 9*: .

Therefore, the operations + and - can be defined on the quotient set defined by the equivalence relation

*Lemma 10*: The binary operation + acting upon the quotient set is an Abelian group with - as the inverse function and 0 as the identity.

The set of nimbers is defined as the smallest subcollection containing 0 and containing for every G in the subcollection.

Nimbers are the combinatorial game theoretic analogue of the ordinals and in fact, we can define a function from the ordinals to nimbers.

*Sprague-Grundy Theorem*: Every impartial game is -equivalent to a nimber. See Sprague-Grundy theorem.