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.
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