Matti Pitkanen (matpitka@pcu.helsinki.fi)
Mon, 16 Aug 1999 22:15:47 +0300 (EET DST)
On Mon, 16 Aug 1999, Stephen P. King wrote:
> Dear Matti and Friends,
>
> Did you read the whole paper? Do you understand what Calude's
> "Lexicons" are?
Yes. As far as it is possible to understand things like this except
through formal construction.
>
> 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 lexicons are phantastic concept but I could not follow the
talk about rationals as novelties and almost all reals as lexicons
containing these novelties.
>
> > 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?
>
I have only construction receipe and no real proofs. I am not a
mathematician!
Let us take example: construct formally the product of all
possible finite primes and call it X.
Infinite primes defined by formula
P= mX/s + ns
s= p1...pn is product of finite primes, which are all different
n contains only powers of p1...pn dividing s as factors.
m is integer not divisible by p1,... or pn.
This number is not divisible by any finite prime. Any finite prime
dividing s divides n*s but does not divide nX/s. Any finite prime
not dividing s divides m*X/s but does not divide n*s. Therefore
P mod p >0 for any finite prime.
I have *no* rigorous proof that P could not have infinite primes
as factors. The formula actually generalizes quite a lot.
One can take the resulting finite and infinite primes and
repeat the construction ad infinitum. There are also more general
infinite primes.
Formula has nice physical interpretation.
a) X which is product of all finite primes represents
Dirac sea with all fermionic states labelled by finite primes filled
b) Division by s creates number of holes in this Dirac sea and
corresponds to the state s= p1...pn. Only single fermion is possible
in state since only first powers of pi are possible.
c) m and n which are arbitrary integers except for the constraint
given above. m and n represent many boson states labelled by finite
primes. Since there is one-one correspondence between 'fermions' and
'bosons', infinite primes obtained in this manner correspond to
many particle states of super symmetric QFT!
> > 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?
Imbedding space in M^4_+ degrees of freedom is infinite and
if one extends reals then one must also extend M^4_+! CP_2 is
compact but can contain infinitesimals.
I remember having read that integral calculus does not
generalize to the surreal world. But at least formally p-adic
numbers with infinite p might exist and that this topology is
effectively real topology. This would suggest that infinite p physics
exists and is well defined and might be an extremely effective
approximation to finite p physics for large p since all S-matrix
elements would effectively contain only two lowest powers of p!
For instance for p=M_127 the expansion in powers of p converges
with extremely rapid rate.
> 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]
Cognitive spacetime sheets can have *infinite but bounded* (by infinite
p-adic length scale) temporal duration. These creatures would see us
as infinitesimals. In natural length scale defined by infinite
p-adic length scale they would experience everything finite!
> 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.
Already lexicons are quite wonderful but what if one allows infinitesimal
parts by allowing pinary expansions x= SUM (n) x(n)p^(-n) in
which n can be product of not only finite prime powers but also
powers of infinite primes?
> 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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