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\section{\usemenu{slacpub7205::context::slacpub7205007}{Remark on Integers}}\label{section::slacpub7205007}
It is clear from our comment on measurement accuracy that we will
find it useful to talk about {\it halfintegers} as well as integers. This will
also be useful when we come to talk about angular momenta and other
``noncommuting observables'' in our language. But how much farther
must we go beyond the positive integers? It was Kronecker
who said ``God gave us the integers. All else is the work of man.''
One of our objectives is to keep this extra work to a minimum.
I am certain that the largest string length segment we will need to
construct the quantum numbers needed to analyze currently available data about the
observable universe of physical cosmology and particle physics
is $256$, and that all we need do with such segments is to combine
or compare or reduce them by the operations listed above,
i.e. discrimination, concatenation and inner product.
Using as a basis bitstrings of length $16 W$, I also see how
to represent negative integers, positive and negative imaginary
integers, and complex integer quaternions. Discussion of how far we
need go in that direction, or into using rational fractions
other than ${1\over 2}$ is,
in my opinion, best left until we find a crying need to do so.
In any case, we have to lay considerable groundwork before we do.
For the moment we assume all we need know about the integers is that
\begin{equation}
1+0=1=0+1;\ \ \ \ 1\times 0=0=0\times 1;\ \ \ \ 1+1=2
\end{equation}
that we can iterate the third equation to obtain the counting numbers
up to some largest integer $N$ that we pick in advance as adequate
for the purpose at hand, and that given any integer $n$ so generated
other than $0$ or $N$, any second integer $n'$ will be greater than,
equal to, or less than the first. That is, we assume that the three cases
\begin{equation}
n' > n\ \ \ \ or \ \ \ n'=n\ \ \ \ or \ \ \ n' < n
\end{equation}
are {\it disjoint}. This already implies that we can talk about larger
integers\cite{34}.
I have found that McGoveran's phrase ``naming a largest integer
in advance'', used above, needs be give more structure in my theory.
I assume that all the quantum numbers I need consider
can be obtained using strings of length 256 or less.
If we have $2^{256}$ such strings, we have more than enough
to {\it count}  in the sense of Archimedes {\it Sand Reckoner} 
the number electrons and nuclei (``visible matter'') in
our universe. The mass of the number of nucleons of protonic mass
needed to form these nuclei is considerably less
than current estimates of the ``closure mass''
of our universe, leaving plenty of room for the observed ``dark
matter''. I also believe that considerably less than the
$256!$ orders in which we could combine the $2^{256}$ distinct strings
of length $256$ will suffice
to provide the raw material for a reasonable model of a historical record of both
cosmic evolution and terrestrial biological, social and cultural
evolution. Such a model can be correct without being complete.
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