Superinjunctions are much in the news in Britain at present.
Last year we had the scandalous case of a tanker which delivered a load of toxic waste to an African port without telling the port authorities the nature of the cargo. Many people were harmed as a result. But the owners took out a superinjunction in an attempt to ensure that they were not brought to book for this extraordinary act; we were not supposed even to know that it had taken place. In the end, they probably did themselves more harm than good by this.
I find it scarcely credible, first that a company can do something like this, and second that the law abets them in trying to cover it up.
In passing, note that the toxic waste was produced because the tanker itself had been used as a chemical factory to produce oil by combining industrial waste with strong acid. In other words, rather than simply carrying oil, it was trying to produce oil as well. Anticipating what comes later, I am tempted to call this a “supertanker” (not in the usual sense of the word).
More recently superinjunctions have been used to cover up sexual misdemeanours by prominent people; we are not to know who the people are, or even that the superinjunctions exist. More worryingly, it has been revealed that the issue cannot even be discussed in the House of Commons. Also, incredibly, it seems that this legal instrument gives no privacy to the other person in the case. (One of the prominent people has broken ranks and is revealed to be a journalist, a member of a profession supposedly dedicated to revealing things which others may wish to conceal.)
But my reason for mentioning it here is that there seems to be a whiff of paradox or even contradiction about a superinjunction. I am not a lawyer, and maybe someone will correct me. But suppose that a journalist finds out (to be topical, perhaps by buying information obtained by hacking voicemail accounts) that a misdemeanour has been committed, but is unaware that it is superinjuncted because of the enforced secrecy, and publishes it. What happens? It seems that the only way that the information can be kept secret is that details of the information must be given to all journalists, just the people who are most likely to reveal it!
Is there anything similar in our subject? Let us say that a super-X is something which is an X but also has some property related to the whole class of Xs. (The analogy with supertankers and superinjunctions is admittedly a bit weak.) One thinks of moduli spaces in geometry. Is there anything paradoxical about these? I don’t know enough to say, but since there is a correlation between thinking about moduli spaces and thinking about toposes, it is possible that there may be.
The best example I can come up with is what is usually called the Hypergame, which using my above terminology I will call the Supergame for the rest of this post.
A 2-player game is well-founded if every play of the game terminates after a finite number of moves. Now a play of the Supergame works as follows: Player 1 chooses any well-founded game; the players then play this game, with Player 2 taking the first move.
Is the Supergame well-founded?
Clearly yes, since any play of the Supergame lasts just one move longer than the play of the game chosen in the first move.
Clearly no. For if it were, then it would be possible for Player 1 to choose to play the Supergame, giving Player 2 the first move; Player 2 could then choose the Supergame; and so on, without termination.