Well put Juergen. The question arose as to whether our universe could be or continous. Don't the computable numbers form a continuum; hence even restricting the universe to one we can describe would still allow it to be continuous?
Brent Meeker On Thu, 21 Jun 2001 [EMAIL PROTECTED] wrote: > There has been some confusion regarding the various kinds of infinity. > > There is the rather harmless kind: the countable one. And some say there > is another kind, a strange one, the one associated with the uncountable > continuum, the one whose very existence many deny. > > Do not lump them together. > > The former is accessible by nonhalting computer programs. The latter is not. > > A program that runs forever cannot consume more than countable time steps > and storage cells, adding a new cell whenever it needs one. For example, > countably many steps and cells are sufficient to print all digits of > the real number Pi. Therefore Pi is "computable in the limit." > > But countable time and space resources are NOT sufficient to print all > digits of all random reals in the continuum. In fact, countable resources > are not even sufficient to print all (countably many) digits of a single > real without finite individual description. Unlike Pi, such truly random > reals (and almost all reals are truly random) are NOT computable in the > limit, although all their finite beginnings are. > > Pi has a finite description. Are all infinite objects with finite > descriptions computable in the limit? No. Counter example: Infinite Omega > (the halting probability of a universal Turing machine with random input) > has a finite description, but countable resources are NOT sufficient > to print it, although each finite beginning of Omega is computable in > the limit. > > Algorithmic TOEs are limited to the comparatively few, countably many, > possibly infinite universe histories with finite descriptions. ATOEs deny > the existence of other histories, of histories we cannot even describe > in principle, of histories we cannot reasonably talk about. > > Likewise, ATOEs are restricted to probabilities computable in the limit. > We cannot formally describe other probabilities, and therefore cannot > write reasonable papers about them. This apparently harmless restriction > is the very reason that any complex future (among all the possible > futures compatible with the anthropic principle) necessarily is unlikely. > > Juergen Schmidhuber > > http://www.idsia.ch/~juergen/ > http://www.idsia.ch/~juergen/everything/html.html > http://www.idsia.ch/~juergen/toesv2/ > >
