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 NO(1).] It is not a computable number;
there is no computable function that enumerates its binary expansion.
[See the URL.]
Rob

Rob Warnock <rpw3@rpw3.org>
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