Matthias Blume  <blume@hana.uchicago.edu> wrote:
+---------------
| "John Thingstad" <jpthing@online.no> writes:
| > Well all irrational numbers and particularly all transcendental numbers.
| 
| False. sqrt(2) is irrational -- and I just represented it. Many
| transcendentals are similarly representable as solutions to certain
| equations (just not algebraic ones). Any number you can describe to me
| unambiguously is representable. And -- by that definition -- you cannot
| unambiguously describe to me a particular number that is not
| representable.
+---------------

Hah! What about numbers you can describe unambiguously but not
compute more than a tiny fraction of their leading bits?!?
[...at least, not to better precision than the length of the
program that's doing the computation.] I refer, of course, to
Chaitin's "Omega" <http://en.wikipedia.org/wiki/Chaitin%27s_constant>,
which is a real number between 0 & 1 but is algorithmically random.
[That is, the shortest program to output the first N bits of Omega
must be of size at least N-O(1).] It is not a computable number;
there is no computable function that enumerates its binary expansion.
[See the URL.]


-Rob

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Rob Warnock			<rpw3@rpw3.org>
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