# Re: Hole in a box

From: Anders Sandberg (asa@nada.kth.se)
Date: Tue Feb 19 2002 - 08:01:13 MST

On Tue, Feb 19, 2002 at 07:32:44AM -0500, Louis Newstrom wrote:
>
> > Anders wrote:
> > > Assume we can store x bits / kg of matter. One stored bit is
> > > worth c^2/x J of energy if its substrate is transmuted.
>
> I disagree with your premise. As Godel pointed out, a single integer can
> store an infinite amount of information:
> Digitize the information into discrete bytes a, b, c, ...
> Encode a single integer as 2^a * 3 ^b * 5^c ... using prime numbers for the
> base.
> Any unique infinate data will have a unique result.
> Then take only TWO atoms and store the number as say the number of inches
> betwen them.
>
> Thus, any two atoms can store an infinite amount of information. (The only
> limit in the scheme I have given is the size of the universe in inches.)
> Given cleverness in how to store large integers, I don't think there is a
> limit based on the amount of matter in the universe.

You can only move slower than c. Storage and retrieval of data will
take exponentially increasing time as it gets longer.

Storing a one megabyte text would require a distance X where log X =
127*(log(2)+log(3)+...+log(1e6/(log(1e6)-1))) ~= 127 sum_1^7.803e4
log(n) ~= 1.0172e8 (here I used the prime number density approximation
pi(n)=n/(log(n)-1)). exp(1e8) is a *very* large number. For
comparision, ln [100 billion lightyears] ~= 64.417. So that megabyte
text is far too large to be encoded in the visible universe. If you
settle for a 100 billion lightyear distance, then you can't use any
character higher than 92 (backslash in ascii) if you want to fit it in.

G�del numbers are great in mathematics, but useless in megascale
computation.

> The real energy shortage will be in how much energy it takes to read/write
> information. All reading and writing of data seems to require energy.
> Thats where the energy cost will lie.

Sure, it is the kTln2 terms I have been using all along.

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Anders Sandberg                                      Towards Ascension!