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

*Tue, 17 Aug 1999 09:04:41 +0300 (EET DST)*

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Dear Stephen et all,

the article about Zeno's paradox was really stimulating.

Some additional comments below.

1. In TGD context quantum jump Einsteinian solution to Zeno paradox

holds is modified somewhat. With respect to geometric time

there is no motion: tortoise becomes 4-dimensional

geometric object. With respect to subjective time

the observed motion of tortoise is discretized with

average time increment of about 10^4 Planck times per quantum

jump: cognitive spacetime sheet jumps by this temporal distance in

each quantum jump and sees new t=constant section of 4-dimensional

tortoise (in good approximation).

2. I only now realized that every infinite prime, whose inverse is

infinitesimal is smaller than 1/0, the largest possible infinity.

3. The concept of lexicon is phantastic but I could not understand

the notion of rational as novelty and subsequent claim that motion

is illusion.

**Comment: The finite rationals in the lexicon could code the information

about quantum jumps as experienced by finite p self: pinary cutoff

would make everything rational. Since almost every real is lexicon one

could conclude that infinite sequences of quantum jumps experienced

by beings characterized by infinite primes would give complete

statistical description of the subjective universe: all possible quantum

jumps would appear infinitely many times in the sequence!

**Comment: Rationals are crucial in the construction of lexicon. If

one allows infinite primes one must allow rationals which are ratios

of infinite integers. This obviously would generalized that idea about

rationals as numbers characterizing sensory experience. In particular the

phase preserving canonical identification playing key role in quantum TGD

would generalize: Pythagorean triangles characterized by infinite

integers would appear.

**Comment: Could one interpret this by saying that at higher levels of

existence (infinite primes) it is *generalized rationals*, which are

directly accessible to sensory experience? Does the theorem about lexicons

generalize to the context of generalized reals in the sense that all

generalized rationals appear infinitely many times in the expansion of

a generic generalized real?

**Question: We perform something like 10^(49) quantum jumps during

lifetime. Could it be really possible to construct the number theory

and quantum physics on basis of this tiny subjective experience (actually

actual wake-up periods are of order few seconds). Could it be

that we actually entangle now and then (during sleep) with spacetime

surfaces having infinite value of p and thus infinite subjective and

geometric memory and remember what happened as funny ideas about infinite

primes and things like that?

**Comment: Ramajunan told that he was in contact with some God(dess?)

during sleep and got his miraculous number theoretic formulas

from this Godd(dess?). Could it be that some higher level

being indeed friendly drilled a small hole in his/(her?) cognitive

spacetime sheet of infinite but bounded duration for

Ramajunan to get attached to its boundary by join along boundaries bond

and get mathematically enlightened?(;-)

******

4. I realized a nice manner to represent surreals (or whatever TGD

version about extension of reals is). Consider definition of a finite

real as pinary expansion:

x= SUM(n>n0) x(n)p^(-n)

a) For ordinary reals all *finite* integers n

are present in the series.

b) For extened reals also *infinite* integers

n are present. Certainly infinite values of n correspond to infinitesimal

contributions in the expansion of x in negative powers of p.

c) How should one define the part of expansion for which the values of n

are infinite? One can make the expansion unique by following trick: sum

over all n expressible as products of finite and infinite primes!

If one can construct *all* infinite primes (I have constructed quite

many good candidates!) one can make sense of this expansion

at least formally.

****

**Comment: One can write the expansion of finite generalized real in the

form

x= xa+xb

xa= SUM x(n)P^(-n)

xb= SUM(N,n) xc(n)p^(-n*N)

where N is infinite integer having *only* infinite primes in its

product decomposition to powers of primes. xa is the finite part

and xb is the infinitesimal part. I would guess that competing

generalizations of reals do not allow this kind of explicit representation

of generalized reals: for instance, surreals are defined with the help

of some kind of intervals. This representation in principle makes

it possible to multiply and sum formally infinite reals (I hope

that no one asks me to do it!).

**Comment: An interesting question is are these extensions *independent

of prime p as base*? Could it be that extensions are mathematically

nonequivalent so that every prime would give rise to slightly different

reals? Could this be the ultimate reason for why p-adics are so

fundamental?

**Comment: The existence of p-adic pseudoconstants, functions having

vanishing p-adic derivative, is a purely

p-adic phenomenon. If one however allows in real context infinitesimals

then all analytic functions of the form

f(x) = g(xa),

where xa is the finite part of x and g is analytic function,

have vanishing derivative with respect to extended reals. With respect

to ordinary reals derivative is nonvanishing. In fact, one can construct

infinite hieararchy of functions of this kind by cutting off xb in

suitable manner.

**Comment: For surreals integral calculus does not exist: am I correct. I

do not however see any obvious reason for why the integral calculus could

not exist for functions defined in the proposed manner. These numbers

are well ordered: this is crucial. The concept of integral function

defines definite integral and one can construct integration theory for

piecewise analytic functions. Apart from infinitesimals integral is

completely determined by f(xa): is this true?

5. Riemann zeta function contains product over factors over

all primes. An interesting question is whether one could understand

something about zeta function by allowing

also infinite primes in the product formula

Z(s) = Prod(p prime) [1/(1-p^s)].

Best,

Matti Pitkanen

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