The game of MineField

MineField is a competition of who can name the largest infinite number.

The rules

MineField is played by two parties, Alice and Bob, who alternate turns. The game ends when one player challenges a turn by the other.

Initially, the players start with an axiomatic basis of set theory. This could be any of the common bases which are generally accepted as consistent.

On her turn, Alice names two things. First, an axiom or axioms which are added cumulatively to the shared logical system, and second, an infinite number. Bob may then either take a turn of his own or challenge. He can challenge in one of two ways.

Challenge type #1 - Bob proves an inconsistency. Remember that axioms collect cumulatively, so he may use any axioms which he or Alice named previously. It isn't even required that he use any of the axioms she just named (see the explanation of rebuttals below.) In this case, Alice has hit one of the many land mines which litter the mathematical landscape.

Challenge type #2 - Bob constructs a number which he proves to be greater than or equal to Alice's without using any of her new axioms (he may use any previously named ones.) This is to prevent cheap strategies like always naming the power set of the previously named infinite.

After Bob successfully challenges, Alice may issue a rebuttal, which is a challenge of an earlier turn of his. This is to prevent him from introducing an inconsistent axiom on his own turn and then refuting his own axioms rather than hers. Bob may rebut Alice's rebuttal by challenging a turn of hers which happened before the one of his she challenged. The chain of rebuttals continues until someone has the last word. That player is the winner.

MineField is an exceedingly poorly behaved game, but that's what makes it interesting. We have no theory of MineField. Any theory of MineField would be a theory of the mathematician's mind.

by Bram Cohen May 2000