## Friday, November 21, 2008

Upon reflection, I don’t think much of my previous resolution of Richards’ Paradox, which was as follows: Such finite strings of words as specify real numbers between 0 and 1 can be listed in order of increasing number of symbols used in the description and then, within each length of description, in alphabetical order. Given that list, one may seem able to define a real number between 0 and 1 and not in that list using a diagonal procedure: The digit in its nth decimal place is the final digit of d + 1, where d is the digit in the nth decimal place of the nth number on the list. Richards’ paradox is that such a number has such a finite description (so it is in our list, whence that procedure is contradictory, where it is, so there is no such number, in our list, whence that procedure is not contradictory, and so on).
......My previous resolution was just to note that our list includes anything that anyone could possible say (in finitely many words) that would specify a real number between 0 and 1. So in order to specify a number via that list, any (finitely describable) diagonal procedure must explicitly exclude its own entries in the list. But that now seems patently inadequate. Given such a list, we have its diagonal, and so a real number not on that list is clearly indicated. And we can say that it is. What we say cannot be in the list, but that just means that such a list—of all the finite specifications of such real numbers—is impossible. I would not be surprised if it was impossible, because words are typically a bit vague and variable in meaning. But it does seem odd that we can deduce that they must be fuzzy or incomplete, which is why I am not very fond of that resolution either. All in all, I find Richards’ paradox very puzzling.

ccima said...

The list of all finite strings as a whole has a diagonal thats NOT a finite string.

Enigman said...

If by "finite string" you mean "finitely specifiable endless sequence of digits" then I think I see your point, ccima. The number obtained from the diagonal procedure (not the diagonal itself of course) is different to each entry on the list, which is ex hypothesi ALL the finitely specifiable ones. However, the problem is that that endless sequence of digits certainly is finitely specifiable. There is a short and sweet specification of the whole list, and a short and sweet specification of the diagonal procedure. That's one way of looking at the paradox: the "diagonal" (so to speak) both is not finitely specifiable, and also is, according to two compelling reasons. (Or did you something else?)

ccima said...

Yes I now see that what I said does not have any bearing on the problem.

Is it possible to understand it like this?

Finite specification + Machine M <--> Finite String

(Finite string --> Real number) --> List L of real numbers

The diagonal is a real number not in L.

But it seams intuitive that such finite specification can exist on some machine M.

A specification running all other possible specifications to generate the diagonal. (Halting at a finite point the finite diagonal can be seen as an equivalence class of real numbers.) Let assume that running all specifications generates the diagonal. If the specification of this real is on place N in the list it differ from itself at position N.

Actually when this diagonal-generating-specification is running it has to run itself to generate the digit at the position N. It is going to look at this position d_NN which is empty.. so either we make it work by choosing some arbitary digit, making it the same as the diagonal. Or otherwise it must not be on a finite position in the list.

Can there be more general specifications schemes avoiding this "computer" analogy?