# Re: Bruno's mathematical reality

`On Sun, Dec 29, 2013 at 2:25 PM, meekerdb <meeke...@verizon.net> wrote:`
```
>  On 12/29/2013 5:56 AM, Bruno Marchal wrote:
>
>
>  On 28 Dec 2013, at 22:23, meekerdb wrote:
>
>  On 12/28/2013 4:09 AM, Bruno Marchal wrote:
>
> For a long time I got opponent saying that we cannot generate
> computationally a random number, and that is right, if we want generate
> only that numbers. but a simple counting algorithm generating all numbers,
> 0, 1, 2, .... 6999500235148668, ... generates all random finite
> incompressible strings,
>
>
> How can a finite string be incompressible?  6999500235148668 in base
> 6999500235148669 is just 10.
>
>
>
>  You can define a finite string as incompressible when the shorter
> combinators to generate it is as lengthy as the string itself.
> This definition is not universal for a finite amount of short sequences
> which indeed will depend of the language used (here combinators).
>
>  Then you can show that such a definition can be made universal by adding
> some constant, which will depend of the universal language.
>
>  It can be shown that most (finite!) numbers, written in any base, are
> random in that sense.
>
>  Of course, 10 is a sort of compression of any string X in some base, but
> if you allow change of base, you will need to send the base with the number
> in the message. If you fix the base, then indeed 10 will be a compression
> of that particular number base, for that language, and it is part of
> incompressibility theory that no definition exist working for all (small)
> numbers.
>
>
> Since all finite numbers are small, I think this means the theory only
> holds in the limit.
>
> Brent
>

Brent,

It is easy to see with the pigeon hole principal.  There are more 2 digit
numbers than 1 digit numbers, and more 3 digit numbers than 2 digit
numbers, and so on.  For any string you can represent using a shorter
string, another "shorter string" must necessarily be displaced.  You can't
keep replacing things with shorter strings because there aren't enough of
them, so as a side-effect, every compression strategy must represent some
strings by larger ones.  In fact, the average size of all possible
compressed messages (with some upper-bound length n) can never be smaller
than the average size of all uncompressed messages.

The only reason compression algorithms are useful is because they are
tailored to represent some class of messages with shorter strings, while
making (the vast majority of) other messages slightly larger.

Jason

>
>  Each particular language will have some exception on the
> incompressibility issue. That should be part of the role of the variable
> constant in the general universal definition.
>
>  Bruno
>
>
>
>
>    http://iridia.ulb.ac.be/~marchal/
>
>
>
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