Inspection to infinity

Has the UK Government found the resources to go beyond the Russell Paradox?

"Suppose that every public library has to compile a catalog of all its books. The catalog is itself one of the library's books, but while some librarians include it in the catalog for completeness, others leave it out, as being self-evident.

Now imagine that all these catalogs are sent to the national library. Some of them include themselves in their listings, others do not. The national librarian compiles two master catalogs - one of all the catalogs that list themselves, and one of all those that don't.

The question is now, should these catalogs list themselves? The 'Catalog of all catalogs that list themselves' is no problem. If the librarian doesn't include it in its own listing, it is still a true catalog of those catalogs that do include themselves. If he does include it, it remains a true catalog of those that list themselves.

However, just as the librarian cannot go wrong with the first master catalog, he is doomed to fail with the second. When it comes to the 'Catalog of all catalogs that don't list themselves', the librarian cannot include it in its own listing, because then it would belong in the other catalog, that of catalogs that do include themselves. However, if the librarian leaves it out, the catalog is incomplete. Either way, it can never be a true catalog of catalogs that do not list themselves."

Today our Department had a Service Excellence Inspector in doing what governmental inspectors do. More interesingly, there was another inspector inspecting the inspector. I couldnt resist asking the inspector inspector whether he recognised the Russell problem and whether he was subsequently inspected by another level of inspector above him. He happily reported that yes he was inspected by someone above and as far as he knew they were also inspected by someone; after that, he thought it shifted to a intergovernmental agency which inspected that level and on and on.

In computability theory, this is called the halting problem: which can be stated as follows: given a description of a program, decide whether the program finishes running or will run forever. So it would appear that inspection goes on forever.