Saturday, July 02, 2005

Oh! To live forever!

I buy books at a constant rate, faster than I can read them. This has the unfortunate consequence that no matter how long I live I can't possibly read all of the books I buy. In fact, the longer I live, the more unread books I will have upon my death. But if I live infinitely long I can read them all.

If I live forever then the set of books I buy will be infinite. So you might think there is a 'race' going on. I'll buy infinitely many books, and read infinitely many, so depending on the relative rates I might buy more than I read or vice versa. But suppose it takes time t_i to read book i. Then I'll have read book i by time t_1+...+t_i. So every book is eventually read. Even if every book contains a bibliography with 100 new books, and I add those books to my to-buy list each time I finish a book, I'll still read them all. Even if I read books slower and slower so book i+1 takes twice as long to read as book i, I'll still get to finish all of my books.

Now let's kick it up a notch. Each book contains an countably infinite bibliography and our goal is to ensure that for every book we read, we eventually also read the books in its bibliography. Here it gets a little harder. If we just take the entire bibliography from the first book and stick it in our to-read list we'll never read any of the books in the bibliographies of those books, we'll always be stuck on the first bibliography. What strategy can we use to achieve our goal? (It can be achieved.)

Anyway, this was derived from Smullyan's arguments about Hintikka sets in his First Order Logic.


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