tl;dr version
He constructs a weird order on the rational numbers, and they play the game on that ordering. He also has steps for limit ordinals.
For example:
Cheryl: I have given each of you an ordinal. I didn't give both of you the same ordinal. Do you know if you have the smallest one?
Bob: I can't prove I have the smallest. (I.e., I don't have 0.)
Albert: With that information, I can't prove I have the smallest. (I.e., I don't have 1.)
Bob: I don't have 2.
Cheryl: You can go on forever without figuring out who has smaller.
Bob: I don't have the smallest. (I.e., I don't have omega.)
Albert: I don't have the smallest. (I.e., I don't have omega+1).
Cheryl: You can do this 100 times and it wouldn't help.
Bob: I don't have the smallest. (I.e., I don't have 100 * omega.)
Albert: I know I have the smallest now! (I.e., I have 100 * omega+1)
The weird ordering he places on the rational numbers is equivalent to playing this way with ordinals.
He constructs a weird order on the rational numbers, and they play the game on that ordering. He also has steps for limit ordinals.
For example:
Cheryl: I have given each of you an ordinal. I didn't give both of you the same ordinal. Do you know if you have the smallest one?
Bob: I can't prove I have the smallest. (I.e., I don't have 0.)
Albert: With that information, I can't prove I have the smallest. (I.e., I don't have 1.)
Bob: I don't have 2.
Cheryl: You can go on forever without figuring out who has smaller.
Bob: I don't have the smallest. (I.e., I don't have omega.)
Albert: I don't have the smallest. (I.e., I don't have omega+1).
Cheryl: You can do this 100 times and it wouldn't help.
Bob: I don't have the smallest. (I.e., I don't have 100 * omega.)
Albert: I know I have the smallest now! (I.e., I have 100 * omega+1)
The weird ordering he places on the rational numbers is equivalent to playing this way with ordinals.