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, including >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 0.111 > 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 representations. Nick Bostrom Dept. Philosophy, Logic and Scientific Method London School of Economics Email: [EMAIL PROTECTED] <mailto:[EMAIL PROTECTED]> Homepage: http://www.analytic.org <http://www.analytic.org>