Stephen P. King (stephenk1@home.com)
Mon, 16 Aug 1999 14:04:49 -0400
Dear Matti and Friends,
Did you read the whole paper? Do you understand what Calude's
"Lexicons" are?
Matti Pitkanen wrote:
>
> On Mon, 16 Aug 1999, Stephen P. King wrote:
>
> > Dear Matti,
> >
> > Please read this paper!
> >
> > http://www.cs.auckland.ac.nz/CDMTCS//researchreports/089walter.pdf
>
> The paper seems that it is about Zeno paradox. I think
> that I have read some popular article about solution of Zeno paradox
> in terms of infitesimal numbers for year or so ago.
Zeno's paradox is the framing of the problem... I am not saying that I
agree with Calude's conclusion, I am just pointing it out. I have a
surprise, but you need to understand What Dr. Meyerstein is saying about
Chris Calude's result to understand a notion that I have been working on
for a very long time! :-)
> The introduction of infinite primes forces automatically also the
> introduction of infinitesimals. All predictions of the
> p-adicized quantum TGD for infinite p:s would be series containing also
> powers of infinitesimals and only the finite part is interesting from our
> point of view: two lowest orders in pinary expansion would give the exact
> result.
What are the axioms that define infinite primes? Can we think of them
as additional postulates for the set theory (Frakel-Zernelo (spelling?))
theory?
> It seems that the testing of our theories with accuracy of
> infinitesimals is a rather remote possibility: and perhaps un-necessary:
> we cannot even agree on basic philosophy! Perhaps those God like
> creatures in the hierarchy of selves, which are labelled by infinite
> primes, are busily constructing physical models in accuracy
> O((1/infinite P)^n) and performing the needed high resolution
> experimentation and reporting various errors using infinitesimals(;-).
What space-time do you think that these creatures exist "in" or do
they, as I suggest, generate their space-times by the very act of
constructing models and performing experiments? What determines the
material structure of the "matter" (and energy) involved? Remember, a
space-time is, literally, an empty and meaningless notion independent of
Local Systems or observer! [quotes are from the paper]
The "lexicon" numbers "Any $finite$ sequence can be unambiguously
coded in binary (or decimal) and thus corresponds exactly to some
rational number"... "on the other hand, real numbers are infinite
sequences of digits (in whatever chosen code or $base$)" "Is there a
real number that with certainty contains the word w? ... Yes ... and
there exists a real number that contains $every possible "word"$. That
is, that contains $everything that can be explicitly stated, coded,
communicated$. ... It can be shown that this special number not only
contains, by construction, every possible finite linear sequence, say
William Shakespeare's complete works, but also that it contains every
possible linear sequence $infinitelt many times$!"
Calude and Zamfirescu have shown that there "exist real numbers that
represent this remarkable property $independent of the employed code or
alphabet$ (binary, decimal, or, for instance, all the symbols on a
computer keyboard). These are the Lexicons. ... The amazing result is:
almost every real number is, both geometrically and
measure-theoretically, a Lexicon! In particular, if you put al the reals
in an urn, and blindly pick one, with almost certainty it will be a
Lexicon."
I see these Lexicons as encoding descriptions of material systems, e.g.
what Local Systems "observe", to be specific! The trick I see is that if
we consider that for every finite sequence there exists a configuration
of matter (in a finitely bounded or closed space-time!) such that the
finite sequence or "word" describes it, given some code or base.
We then ask: By what procedure are "configurations of matter" matched
up with "words" such that their "meaning" can be communicated and
decoded by another LS?
Let us take a long hard look at what Pratt is telling us!
Onward!
Stephen
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