**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Mon, 16 Aug 1999 22:15:47 +0300 (EET DST)*

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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
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*> > creatures in the hierarchy of selves, which are labelled by infinite
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*> > 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
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*> coded in binary (or decimal) and thus corresponds exactly to some
*

*> rational number"... "on the other hand, real numbers are infinite
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*> sequences of digits (in whatever chosen code or $base$)" "Is there a
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*> real number that with certainty contains the word w? ... Yes ... and
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*> there exists a real number that contains $every possible "word"$. That
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*> is, that contains $everything that can be explicitly stated, coded,
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*> communicated$. ... It can be shown that this special number not only
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*> contains, by construction, every possible finite linear sequence, say
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*> William Shakespeare's complete works, but also that it contains every
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*> possible linear sequence $infinitelt many times$!"
*

*> Calude and Zamfirescu have shown that there "exist real numbers that
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*> represent this remarkable property $independent of the employed code or
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*> alphabet$ (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
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*> measure-theoretically, a Lexicon! In particular, if you put al the reals
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*> 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
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*> of matter (in a finitely bounded or closed space-time!) such that the
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*> 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
*

*>
*

**Next message:**Stephen P. King: "[time 558] Re: [time 556] Modeling change with nonstandardnumbers"**Previous message:**Matti Pitkanen: "[time 556] Re: [time 553] Re: [time 535]Modeling change with nonstandard numbers"**In reply to:**Matti Pitkanen: "[time 553] Re: [time 535]Modeling change with nonstandard numbers"**Next in thread:**Matti Pitkanen: "[time 556] Re: [time 553] Re: [time 535]Modeling change with nonstandard numbers"

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