### Hyperfun!

Smullyan's favourite game is called the

Consider games where the players take turns and the game is guaranteed to terminate in a finite number of moves. These are call

The Hypergame is quite simple. The first player chooses a well-founded game. The game now turns into that game and the second player starts by opening in that game with play continuing in that game. Eg. if the first player says "Chess (with whatever house rules are required to make it terminate)" then they start playing Chess with the second player moving first. Because of the stipulation of well-foundedness the Hypergame lasts precisely one move more than some well-founded game. Therefore the Hypergame is itself well-founded and always terminates.

Anyway, Alice and Bob decide to play:

Alice: Let's play the Hypergame!

Bob: Cool! I'll go first. My move is to select the Hypergame.

Alice: OK, now we're playing the hypergame. So my first move is to select the Hypergame.

Bob: The Hypergame is cool, I pick that.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

etc.

*Hypergame*, invented by Zwicker. I hadn't heard of it until today when I read about it in Barwise and Moss's book*Vicious Circles*.Consider games where the players take turns and the game is guaranteed to terminate in a finite number of moves. These are call

*well-founded*games.The Hypergame is quite simple. The first player chooses a well-founded game. The game now turns into that game and the second player starts by opening in that game with play continuing in that game. Eg. if the first player says "Chess (with whatever house rules are required to make it terminate)" then they start playing Chess with the second player moving first. Because of the stipulation of well-foundedness the Hypergame lasts precisely one move more than some well-founded game. Therefore the Hypergame is itself well-founded and always terminates.

Anyway, Alice and Bob decide to play:

Alice: Let's play the Hypergame!

Bob: Cool! I'll go first. My move is to select the Hypergame.

Alice: OK, now we're playing the hypergame. So my first move is to select the Hypergame.

Bob: The Hypergame is cool, I pick that.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

Alice: I pick the Hypergame.

Bob: I pick the Hypergame.

etc.

Labels: mathematics

## 2 Comments:

Nice paradox!

I suppose to block this we have to say that a game is a pair of

setsof games, rather thanclasses- and since there is not set of all well-founded games, but only a proper class, the move is blocked."Vicious Circles" discusses your 'block' and rejects it as unsatisfying. But I haven't read the bit where it proposes its own resolution - presumably based on the anti-foundation axiom, which I haven't read about yet either.

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