Bruno wrote:

                >Juergen Schmidhuber:

                >> Even the set of all real #'s is not formally describable.
You cannot
                >> write a program that lists all reals.  In infinite but
countable time
                >> you can write down a particular computable real, but not
all reals.

                >I don't agree. Indeed, you cannot write a program that
lists all reals.
                >But in infinite and still countable time you can write down
all the reals,
                >by dovetailing on all initial segment of all the reals,
                >non computable one.

                 >                 0.0                           0.1
                 >           0.00       0.01              0.10         0.11
                 >      0.000  0.001 0.010  0.011    0.100  0.101   0.110
                 >                etc.                         etc.

                >There is of course no contradiction with the fact that most
of the reals
                >are uncomputable, and that the set of reals is uncountable.

No you've got that wrong, Bruno. The output of the process you describe is a
countable string of reals, and it thus leaves out the vast majority of all
real numbers. (To see that the string is countable, map it to the integers
as follows: upper-left=1, upper-right = 2, second row-first column = 3 etc.;
this mapping will include all entries in your listing) -- What you are
leaving out are a lot of reals that have an infinite binary decimal

Nick Bostrom
Dept. Philosophy, Logic and Scientific Method
London School of Economics
Homepage: <> 

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