Non-standard gap


Here's a game, entitled « non-standard gap », which lets you figure out if the person you're speaking to has a non-standard mind, or - and surely it's the same thing - if they're secretely a set theorist.

This is a two-player game where Alice and Bob play in turns. At the -th round, Alice sets forth an ordinal , then Bob proposes an ordinal , with the following constraints:


Alice loses the game in the -th round if Bob submits her a winning strategy, with proof that in less than a determined number of rounds after this -th one, Alice won't be able to extend her sequence while observing the constraints (that is, we'll have for a certain ). Bob loses the game if he gives up or dies of thirst.

Example. Alice plays the number ;

Bob plays ;

Alice plays ;

Bob plays ;

Alice plays ;

Bob plays ;

Alice plays .

Bob then proposes the following winning strategy:

If Alice has just played the number

where , then Bob will play

where is the predecessor of in for the appropriate lexicographical ordering. Bob claims, with proof, that Alice will lose in at most additional rounds.

Question 1. Does Bob have a winning strategy? The ordinal being fixed, does Bob have a winning strategy? If not, then which is this ordinal that is smallest, for which Bob has no winning strategy?

Question 2. Does Alice have a winning strategy?

Remark. I never lost a game of non-standard gap.