[time 767] Generalizing the concept of integer, rational and real, etc..

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 16 Sep 1999 10:38:21 +0300 (EET DST)

The discussions with Stephen about lexicons inspired the attempt
to define precisely the concepts of generalized integer, rational
and real. Rather surprisingly, infinite integers and generalized
reals can be regarded as infinite-dimensional vectors spaces
having ordinary integers and reals as coefficients. Thus
the tangent space of configuration space of 3-surfaces might
perhaps be regarded as the space of generalized octonions!

The concept of generalized lexicon lends support for the idea
that TGD:eish Universe has enough subjective memory to discover
laws of physics and consciousness.

'Infinite primes and consciousnes' in 'TGD inspire theory
of consciousness...' at
http://www.physics.helsinki.fi/~matpitka/cbook.html

Best,
MP
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\newcommand{\vm}{\vspace{0.2cm}}
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\subsection{How to generalize the concepts of integer,
rational and real?}

The allowance of infinite primes forces to generalize also the concepts
concepts of integer, rational and real number. It is not
obvious how this could be achieved. The following arguments
lead to a possible generalization which seems practical (yes!) and
elegant.

\subsubsection{Infinite integers form infinite-dimensional vector space
with integer coefficients}

The first guess is that infinite integers $N$ could be
defined as products of the powers of finite and infinite primes.

\begin{eqnarray}
N&=&\prod_k p_k^{n_k}= nM\per , \per n_k\geq 0\per ,
\label{product}
\end{eqnarray}

\noindent where $n$ is finite integer and $M$ is infinite integer
containing only powers of infinite primes in its product expansion.

\vm

It is not however not clear whether the sums of infinite integers
really allow similar decomposition. Even in the case
that this decomposition exists, there seems to be no way of
deriving it. This would suggest that one
should regard sums

$$\sum_i n_iM_i$$

\noindent of infinite integers
as infinite-dimensional linear space spanned by $M_i$
so that the set of infinite integers would be analogous
to an infinite-dimensional algebraic extension of say p-adic numbers
such that each coordinate axes in the extension corresponds to single
infinite integer of form $N=mM$.
Thus the most general infinite integer $N$ would have the form

\begin{eqnarray}
N&=& m_0+ \sum m_iM_i\per .
\end{eqnarray}

\noindent This representation of infinite integers
indeed looks promising from the point of view of
practical calculations. The representation looks also
attractive physically.
The integers $m_0$ and $m_iM_i$ are formally analogous to a many-boson
states consisting of $n_k$ bosons in the mode characterized by finite
or infinite prime $p_k$. Therefore this
representation is analogous to a quantum superposition
of bosonic Fock states with binary, rather than complex valued,
superposition coefficients.

\subsubsection{Generalized rationals}

Generalized rationals could be defined
as ratios $R=M/N$ of the generalized integers. This works nicely
when $M$ and $N$ are expressible as products of powers of finite
or infinite primes but for more general integers the definition
does not look attractive. This suggests that one should restrict
the generalized rationals to be numbers having the expansion
as a product of positive and negative primes, finite or infinite:

\begin{eqnarray}
N&=&\prod_k p_k^{n_k}= \frac{n_1M_1}{nM}\per .
\label{rational}
\end{eqnarray}

\subsubsection{Generalized reals form infinite-dimensional
real vector space}

One could consider the possibility of defining
generalized reals as limiting values of the generalized rationals.
A more practical definition of the generalized reals is
based on the generalization of the pinary expansion of ordinary
real number given by

\begin{eqnarray}
x&=& \sum_{n\geq n_0} x_np^{-n}\per ,\nonumber\\
x_n&\in& \{0,..,p-1\} \per .
\end{eqnarray}

\noindent It is natural to try to generalize this expansion
somehow. The natural requirement is that
sums and products of the generalized reals
and canonical identification map from the generalized
Only in this manner the representation can
have practical value.

\vm

These requirements suggest the following generalization

\begin{eqnarray}
X&=& x_0+ \sum_{N} x_N p^{-N}\per ,\nonumber\\
\end{eqnarray}

\noindent where $x_0$ and $x_N$ are ordinary reals, $N$ denotes
infinite integer.
Note that generalized reals can be regarded as infinite-dimensional
linear space such that each infinite integer $N$ corresponds
to one coordinate axis of this space.
The sum of two generalized reals can be
readily calculated by using only sum for reals:

\begin{eqnarray}
X+Y&=& x_0+y_0 + \sum_{N} (x_N +y_N)p^{-N}\per ,\nonumber\\
\end{eqnarray}

\noindent The product $XY$ is expressible in the form

\begin{eqnarray}
XY&=& x_0y_0 +x_0Y+Xy_0 + \sum_{N_1,N_2} x_{N_1} y_{N_2}
p^{-N_1-N_2}\per ,\nonumber\\
\end{eqnarray}

\noindent If one assumes that
infinite integers form infinite-dimensional vector space in the
manner proposed, there are no problems and one can calculate
the sums $N_1+N_2$ by summing component wise manner
the coefficients appearing in the sums defining
$N_1$ and $N_2$ in terms of
infinite integers $M_i$ allowing expression as a product
of infinite integers.

\vm

Canonical identification map from ordinary reals to p-adics

$$x =\sum_k x_kp^{-k}\per \rightarrow \per x_p= \sum_k x_kp^{k}\per ,$$

\noindent generalizes to the form

\begin{eqnarray}
x&=& x_0+ \sum_{N} x_N p^{-N}\per \rightarrow \per
(x_0)_p+ \sum_{N} (x_N)_p p^{N}\per ,
\end{eqnarray}

\noindent so that all the basic requirements
making the concept of generalized real calculationally useful
are satisfied.

\vm

There are several interesting questions related to generalized reals.

a) Are the extensions of
reals defined by various values of p-adic primes mathematically
equivalent or not? One can map
generalized reals associated with various choices of the base $p$
to each other in one-one manner using the mapping

\begin{eqnarray}
X&=& x_0+ \sum_{N} x_N p_1^{-N} \per \rightarrow \per x_0+
\sum_{N} x_N p_2^{-N}\per .\nonumber\\
\end{eqnarray}

\noindent The ordinary real norms of
{\it finite} (this is important!) generalized reals are
identical since the representations
associated with different values of base $p$ differ from each other only
infinitesimally.
This would suggest that the extensions are
physically equivalent.
It these extensions are not mathematically equivalent then p-adic
primes could have a deep role in the definition of the generalized
reals.

b) Could one perhaps regard
infinite-dimensional configuration space of 3-surfaces
or rather, its tangent space, as a realization of the
generalized reals?
Or could one perhaps regard the tangent space of
the configuration space
as infinite-dimensional algebraic extension
of octonions as the 8-dimensionality of the imbedding
space suggests? This kind of identification could perhaps reduce
physics to generalized number theory.

\subsection{Can TGD:eish Universe discover the laws of physics
and consciousness?}

Any theory of consciousness should predict the possibility
of its own discovery. It seems that TGD:eish universe could
indeed discover itself.

\subsubsection{Geometric and subjective memories and self hierarchy}

TGD:eish Universe has the following tools to discover itself.

a) Geometric memory resulting from the fact
that the contents of conscious experience is determined by
the entire initial and final quantum {\it histories},
makes possible conscious simulations of the geometric
time development determined by the deterministic laws
of physics. What results are prophecies of future and
past: that is conscious information about
how universe would evolve in subjective future and
how it would have
evolved in subjective past assuming that quantum jumps had no
effect on the time evolution. Classical physics can be regarded as
a good example of geometric memory.

b) Subjective memory is memories about conscious experiences.
Without subjective memory it would not be possible to even
discover the notion of quantum jump! Thus there seems to
be no reason, which would not allow
TGD:eish Universe to discover also quantum theory and theory
of consciousness.

c) Self hierarchy and summation hypothesis imply an infinite
hierarchy of abstractions and at the top of the hierarchy
is the entire Universe possessing subjective memory
In order to avoid the question What was the initial
quantum history?'', one must assume that the number of these
quantum jumps is infinite. The cardinality
for the number of the quantum jumps occurred could quite
well be larger than the cardinality of natural numbers.
This means that Universe has huge life experience and
it would not be wonder if it could construct rather precise
theory of EveryThing!

\subsubsection{Does a generic quantum jump sequences of the Universe
contain each possible quantum jump infinitely many times?}

A natural question is whether the infinite sequence
of quantum jumps could give complete information
quantum statistical determinism would be exact at
the level of the entire universe and Universe could
be able to discover the precise quantum laws dictating
its behaviour. This would require
that the infinite sequence of quantum jumps
contain every possible quantum jump infinitely many times!
This requirement is probably too strong.
The presence of the pinary cutoff however provides a natural manner to
classify quantum jumps to equivalence classes such that members
in same equivalence class are identical in the accuracy
provided by the pinary cutoff. Hence the question is
whether each possible equivalence class appears infinitely many times
in the sequence: that is, whether universe has experienced all
possible moments of consciousness infinitely many times.

\vm

Pinary cutoff presumably means that each
class of quantum jumps for each self can be coded by a
a finite integer. Thus it would seem
that each quantum jump of the Universe could be characterized
by an infinite sequence of finite integers, one for each self
the time evolution of the Universe could be coded to an
infinite sequence of reals with each real being obtained
by simply lumping the integers characterizing invididual
quantum jumps to a sequence using some convention to distinguish
between the integers characterizing different quantum jumps.
Thus the question has been transformed to the following
form:

\vm

{\it Does each real in this infinite sequence of reals
contain each integer infinitely many times in the generic case?}

\noindent The concept of lexicon introduced by Calude \cite{lexicon,Zeno}
suggests that this might be the case!

\subsubsection{The concept of generalized lexicon does it!}

Lexicons are
real numbers, whose expansion contains infinitely many times
the binary (or pinary) expansion of any finite integer. Furthermore,
almost all reals are actually lexicons
\cite{Calude}! The generalization of the concept real number
suggests a generalization of the concept of lexicon.
Generalized reals correspond to the points of an
infinite-dimensional real vector space with coordinate axes
labelled by infinite integers $N$. Generalized lexicons
correspond to generalized reals such that each
real component of the infinite-dimensional vector characterizing
generalized real, is lexicon. But this kind of number is expected
to describe
the quantum jump sequence associated with the entire universe!
>From the properties of the lexicons it would thus follow that each
equivalence class of possible sub-quantum
jumps characterized by pinary cutoff
appears infinitely many times in the sequence of quantums jumps
of entire Universe. This would
mean that statistical determinism is realized exactly and that TGD:eish
Universe has good changes to discover its the laws of physics
and consciousness.

\vm

\begin{thebibliography}{99}
\bibitem[Calude and Zamifrescu]{lexicon}
C. Calude and T. Zamifrescu (1998), {\em The Typical Number is a Lexicon},
New Zealand Journal of Mathematics, Volu. 27, 7-13.

\bibitem[Meyerstein]{Zeno}
F.W. Meyerstein (1999), {\it Is movement an illusion?: Zeno's paradox
from a modern point of view.}\\
http://www.cs.auckland.ac.nz/CDMTCS//researchreports/089walter.pdf.
\end{thebibliography}

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