But the tape can also hold an encoding of the Turing machine to
perform the interpretation. This is the essence of the "compiler
theorem". One can simply iterate this process such that there is no
"concrete" machine interpreting the tape. I think this is another way
of putting the UDA.


On Fri, Mar 17, 2006 at 01:31:22PM -0800, Norman Samish wrote:
> > "Hal Finney" wrote:
> > The first is that numbers are really far more complex than they seem.
> > When we think of numbers, we tend to think of simple ones, like 2, or 7.
> > But they are not really typical of numbers.  Even restricting ourselves to
> > the integers, the information content of the "average" number is enormous;
> > by some reasoning, infinite.  Most numbers are a lot bigger than 2 or 7!
> > They are big enough to hold all of the information in our whole universe;
> > indeed, all of the information in virtually every possible variant of our
> > universe.  A single number can (in some sense) hold this much information.
> How ? Surely this claim needs justification!
> ~~~~~~~~~~~~~~~~~~~~~
> The single number can be of infinite length, with infinite digits, and can 
> therefore contain unlimited information.  One could compare the single number 
> to a tape to a Universal Turing Machine.  Granted, the UTM needs a head and a 
> program to read the tape, so the tape by itself is not sufficient to hold 
> information.
> Norman
> ~~~~~~~~~~~~~~~~`

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A/Prof Russell Standish                  Phone 8308 3119 (mobile)
Mathematics                                    0425 253119 (")
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