The Universe of Discourse

Tue, 10 Jan 2012

Elaborations of Russell's paradox
When Katara was five or six, I told her about Russell's paradox in the following form: in a certain library, some books are catalogs that contain lists of other books. For example, there is a catalog of all the books on the second floor, and a catalog of all the books about birds. Some catalogs might include themselves. For example, the catalog of all the books in the library certainly includes itself. Such catalogs have red covers; the other catalogs, which do not include themselves, such as the catalog of all the plays of Shakespeare, have blue covers. Now is there a catalog of all the catalogs with blue covers?

I wasn't sure she would get this, but it succeeded much better than I expected. After I prompted her to consider what color cover it would have, she thought it out, first ruling out one color, and then, when she got to the second color, she just started laughing.

A couple of days ago she asked me if I could think of anything that was like that but with three different colors. Put on the spot, I suggested she consider what would happen if there could be green catalogs that might or might not include themselves. This is somewhat interesting, because you now can have a catalog of all the blue catalogs; it can have a green cover. But I soon thought of a much better extension.

I gave it to Katara like this: say you have a catalog, let's call it X. If X mentions a catalog that mentions X, it has a gold stripe on the spine. Otherwise, it has a silver stripe. Now:

1. Could there be a red catalog with a gold stripe?
2. Could there be a red catalog with a silver stripe?
3. Could there be a blue catalog with a gold stripe?
4. Could there be a blue catalog with a silver stripe?

And more interesting:

5. Is there a catalog of all the catalogs with gold stripes?
6. Is there a catalog of all the catalogs with silver stripes?

I knew that early 20th century logicians, trying to repair the Russell paradox, first tried a very small patch: since comprehension over the predicate X∉X causes problems, just forbid that predicate. This unfortunately doesn't solve the problem at all; there are an infinite number of equally problematic predicates. (Whitehead and Russell's theory of types is an attempt to fix this; Quine's New Foundations is a different attempt.) One of these predicates is ¬∃Y.X∈Y and Y∈X. You can't construct the set of all X such that ¬∃Y.X∈Y and Y∈X because there is no such set, for reasons similar to the reason why there's no set of all X such that X∉X, so that's where I got the silver stripe predicate.

Translating this into barber language is left as an exercise for the reader.

[ Addendum 20231128: More about uncountable sets for seven-year-olds. ]

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