Can this be shown with an extension of a "pre-fix/don't care bits"
argument? I'm just making this up on the spot, so I'm sticking my neck
out. It's not rigorous, but it could go something like this:
The binary (say) recording of "Gond With The Wind" can be viewed as one
huge but finite binary sequence of n bits:
GWTW = "01001010110 ... binary("Frankly, my dear,...") ... 01001101001".
Actually we can consider all numbers to be an infinitely long binary
sequence, finite ones having an infinite number of leading 0's. So...
1. Of all binary numbers, the probability that the 1st through nth bits
= GWTW is 1/(2^n).
2. So doesn't it follow that, likewise, the probability that the 2nd
through (n+1)th bits = GWTW is 1/(2^n)?
3. So the probability that either #1 or #2 is true is 2/(2^n) =
1/(2^(n-1)). (What about both being true? See step #4.)
4. Now if it weren't for one complication, we would be able to say by
induction the probability that GWTW will be found in the first 2n bits
would be 1 (which obviously is false). The complication is that as we
look at more bits in the pre-fix, there is a relatively small
probability that, depending on the nature of the patterns of 0's and
1's in GWTW, we could find GWTW more than once in the extended prefix.
(e.g. To find GWTW more than once in the first n+1 bits, GWTW would
have to be either all 1's or all 0's.) So we would have to refrain
from counting those certain "multiple occurrence containing" numbers
more than once, slightly decreasing the probability. However, we can
hand-wave and say that this probability is small and thus does not take
us far from a probability of 1 of finding GWTW in a number.
Especially considering all infinitely long numbers, intuitively I'd
guess it's a subset of measure zero that doesn't have a given finite
string inside it somewhere. It might be analogous to saying,"What is
the measure of a subset of the reals that can be described as a
quotient a/b, given that b is fixed." Well we know that is a subset of
the rationals which has measure zero in the reals.
More along the lines of this thread, I'm aware of the weirdnesses in
divergent series, and that it depends on the definition of
divergence/convergence and its context (for instance analytic
continuation). This again gets to the controversial borders of what
mathematics is, and how it relates to reality, and what topology if any
we should choose for Everything (why the complex plane?).
Tom
-----Original Message-----
From: Bruno Marchal <[EMAIL PROTECTED]>
To: everything-list@googlegroups.com
Sent: Sat, 1 Apr 2006 15:47:29 +0200
Subject: Re: The Riemann Zeta Pythagorean TOE
Let us just take the numbers, I mean the finite numbers 0, 1, 2, ...
But let us take them all.
Then it can be shown that numbers without an encoding of "Gone with the
wind" are quite exceptional. Almost all natural numbers, written in
any base, has an encoding of "Gone with the wind", and of the complete
work of Feynman too, and the complete archive of the everything-list.
In the land of big numbers those numbers *who don't* are rare and
exceptional.
It is not entirely obvious. There is a proof of this in the Hardy and
Wright Introduction to Number Theory.
http://www.amazon.com/gp/product/0198531710/103-1630254-7840640?
v=glance&n=283155
Bruno
Le 31-mars-06, à 23:34, [EMAIL PROTECTED] a écrit :
>
> John,
>
> If I understand what you're asking: A digital recording of "Gone With
> The Wind", say on a CD, is recorded in bits, binary digits, 1's and
> 0's. You can also express pi in binary, it's simply the base-2
> representation of pi, all 1's and 0's, just like the movie recording.
> So you have an infinite sequence of 0's and 1's which is the
> representation of pi in which to search for the finite sequence of the
> movie recording.
>
> Tom
>
> -----Original Message-----
> From: John M <[EMAIL PROTECTED]>
> To: everything-list@googlegroups.com
> Sent: Fri, 31 Mar 2006 12:59:20 -0800 (PST)
> Subject: Re: The Riemann Zeta Pythagorean TOE
>
>
> Tom,
>
> may I humblly ask for an example, HOW you would
> imagine the 'sequence' in pi's infinite variety of
> numbers the connotation for "Gone With The Wind - the
> movie?"
> Just 'per apices', show the kind of sequence included,
> I don't want all the details.
>
> Thank you
>
> John M
>
> --- [EMAIL PROTECTED] wrote:
>
>> Interesting! This reminds me of the old standby
>> example of being able to
>> find any sequence of digits in the digits of pi, and
>> therefore being able to
>> find whole digital "recordings" of "Gone With The
>> Wind" or anything you desire,
>> including your-whole-life-as-you-desire-it-to-be, if
>> you search long enough.
>> ;) But that's the key, in my view. It requires
>> desiring, searching and
>> finding. That requires a person. Similarly, it
>> requires a person to combine
>> addition and multiplication. This is because it
>> requires a person to think of
>> grouping things. This is because it takes a person
>> to define meaning.
>>
>> Tom
>>
>> "An equation for me has no meaning unless it
>> expresses a thought of God."
>> Ramanujan
>> "Ask and it will be given to you, seek and you will
>> find, knock and the door
>> will be opened to you." Jesus
>>
>>
>>
>>
>>
>
>
>
>
>
> >
>
http://iridia.ulb.ac.be/~marchal/
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