I believe Chaitin has a definition of randomness that works for finite
strings. If I remember correctly it has to do with the length of the
shortest program that outputs the string being longer than the string
itself.

Jason

On Wed, Mar 10, 2021, 3:43 PM 'Brent Meeker' via Everything List <
everything-list@googlegroups.com> wrote:

>
>
> On 3/10/2021 7:15 AM, Bruno Marchal wrote:
>
>
> On 10 Mar 2021, at 00:19, 'Brent Meeker' via Everything List <
> everything-list@googlegroups.com> wrote:
>
>
>
> On 3/9/2021 2:00 PM, Bruce Kellett wrote:
>
> On Wed, Mar 10, 2021 at 6:00 AM Jason Resch <jasonre...@gmail.com> wrote:
>
>> On Tue, Mar 9, 2021 at 9:41 AM Lawrence Crowell <
>> goldenfieldquaterni...@gmail.com> wrote:
>>
>>> Nothingness is a paradoxical thing. Does nothingness exist? If so, then
>>> by having existential properties it is not pure nothingness. If nothingness
>>> does not exist then there must exist something. In a sense God is the
>>> antithesis of nothingness and in a sense shares the same paradoxical issue.
>>
>>
>> There is a strange and paradoxical sort of identity between *nothing*
>> and *everything*, particularly as it relates to information theory.
>> Insofar as the total set of all possibilities has zero information content.
>>
>
> Even if it tells us what is not possible?  I think you're getting in over
> your head.  What kind of "possible" to you mean?  Simple not
> self-contradictory?  Nomological?  Or what?
>
>
>
> A random message string can contain zero information, but still exist --
> written on a piece of paper, for example.
>
>
> I agree with your basic point, but a random string carries maximum
> information, per Shannon.  That's why maximally compressed string looks
> random; although you can't really define random in the information
> theoretic sense for finite strings.
>
>
>
> You can define randomness for finite strings, up to a constant.
>
>
> What does it mean "up to a constant"?
>
> Most universal machine will agree on some large string being random, but
> can differ on strings shorter than themselves, say. See the book by
> Calllude on the randomness of finite string.
> This is usually defined first, and then an infinite sequence is said to be
> random if almost all his initial segments are.
>
>
> Even with only two "l"s in his name, I find no reference to him.  If you
> have a finite string you can just adopt a notation in which it has a short
> name, "Bob", and then  it's Kolomogorov complexity is that of "Bob".  So I
> don't see by what definition you can prove a finite string to be random.
>
> Brent
>
>
> Bruno
>
>
>
>
>
> Brent
>
> This idea that zero information equates to 'nothing' is just an elementary
> confusion of categories.
>
> This is the main subject of Russell Standish's book: Theory of Nothing:
>> https://www.hpcoders.com.au/nothing.html
>>
>
>
> That is why Russell got so many things wrong in this book.
>
> Bruce
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