-th round, Alice sets forth
an ordinal 
, then Bob
proposes an ordinal 
, with
the following constraints:
|
Késako?
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:
 |
(1) |
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 
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
Remark. I never lost a game of non-standard gap.