Re: URI denumerability

At 13:57 2002 08 28 -0700, Roy T. Fielding wrote:

>>At 06:45 PM 8/26/02 -0400, Ian B. Jacobs wrote:
>>> Resolved: Move "Some resources do not have URIs. URIs are
>>> denumerable, which means there are enough to give one to every
>>> real number without collisions, for example." to footnote.
>>
>>Surely, this is wrong? [1]  (The bit about giving one to every real number.)
>
>Yes, that was wrong -- denumerable would mean they are equivalent to
>the set of natural numbers, not real numbers.  We started off
>discussing that and then fell down a rathole about it not belonging
>in the document at all, and then further into the pits of despair
>when I pointed out that URI are equivalent to real numbers anyway,
>which means they are not denumerable.
>
>In any case, I agree that it should not be in the architecture document.
>
>>[1] http://www.math.utah.edu/~alfeld/math/sets/realproof.html
>
>Just out of curiosity, could someone please explain why that same
>proof cannot be used to prove that URI are not denumerable?  Just
>replace the real numbers in the proof with their equivalent
>representation as a URI.


The proof relies on the fact that the decimal representation of
a real number can have an infinite number of digits.

So the set of URIs is not denumerable if you allow a URI to
have an infinite number of characters in its representation.

paul