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\noindent {\bf DEFINABILITY IN PHYSICS }
\medskip
\noindent D.J. BENDANIEL\\
Cornell University\\
Ithaca NY, 14853,\\
USA
\\
\medskip\noindent \textbf{Abstract} The concept of definability of
physical fields in a set-theoretical foundation is introduced. We
propose an axiomatic set theory and show that the Schr$\ddot o$dinger
equation and, more generally, a nonlinear sigma model can be derived
from a null postulate and that quantization of fields is equivalent to
definability. We find that space-time is relational in this theory.
Some examples of the relevance to physics are suggested.
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\bigskip
\noindent A set U of finite integers is definable in the set theory if
and only if there exists a formula $\Phi_U (n)$ from which we can
unequivocally determine whether a given finite integer n is a member of
U or not.\cite{1} We can extend this concept to physical fields by
asserting
that a physical field in a finite region of space is definable in a
set-theoretical foundation if and only if the set of distributions of
the fields energy among eigenstates can be mirrored in that theory by
a definable set of finite integers. This concept of definability is
appropriate because, were there a field whose set of energy
distributions among eigenstates corresponded to an undefinable set of
finite integers, that field would have at least one energy
distribution whose presence or absence is impossible to determine, so
the field could not be verifiable or falsifiable. Therefore, our task
is to find a foundation in which it is possible to specify the
definable sets of finite integers and which must also contain the
mathematics necessary for the physical fields corresponding to the
sets.
%\begin{table*}
\begin{center}
\begin{tabular}{|l|l|}\hline
Extensionality & Two sets with just the same members are equal.\\
\hline Pairs & For every two sets, there is a set that contains just
them. \\ \hline Union & For every set of sets, there is a set with
just all their\\& members. \\ \hline Infinity & There is at least one
set
$\omega^*$ with members determined in \\& infinite succession. \\
\hline
Power Set & For every set, there is a set containing just all its
subsets. \\ \hline Regularity & Every non-empty set has a minimal
member\\ & (i.e. ``weak'' regularity). \\ \hline Replacement &
Replacing
members of a set one-for-one creates a set.\\ \hline
\end{tabular}
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%\end{table*}
\medskip
\noindent These axioms are the well-known theory of
Zermelo-Frankel (ZF) with the axiom schema of subsets deleted.
As a result of
that deletion, all theorems must hold for every $\omega^*$. The
minimal $\omega^*$, usually called $\omega$, cannot be obtained, so
that both finite and infinite integers exist in $\omega^*$. This
implies that all sets of finite integers are finite and hence
definable.
\medskip
\noindent We can now adjoin to this theory another axiom asserting
that all subsets of $\omega^*$ are constructible. By constructible
sets we mean sets that are generated sequentially by some process, one
after the other, so that the process well-orders the sets. G$\ddot
o$del has shown that an axiom asserting that all sets are
constructible can be consistently added to ZF, giving a theory called
ZFC$^+$.\cite{2} It has also been shown that no more that countably
many
subsets of $\omega^*$ can be proven to exist in ZFC$^+$. Both these
results will, of course, hold for the sub-theory ZFC$^+$ minus the
axiom schema of subsets. Therefore we can adjoin a new axiom
asserting that the subsets of $\omega^*$ are constructible and there
are countably many such subsets. We shall call these eight axioms
Theory T.
\medskip
\noindent We first show that T contains a real line. Recall the
definition of ``rational numbers'' as the set of ratios, usually
called Q, of any two members of the set $\omega$. In T, we can
likewise, using the axiom of unions, establish for $\omega^*$ the set
of ratios of any two of its integers, finite or infinite. This will
be an ``enlargement'' of the rational numbers and we shall call this
enlargement Q$^*$. Two members of Q$^*$ are called ``identical'' if
their ratio is 1. We now employ the symbol ``$\equiv$'' for ``is
identical to''. An ``infinitesimal'' is a member of Q$^*$ ``equal''
to 0, i.e., letting $y$ signify the member and employing the symbol
``$=$'' to signify equality, $y=0 \leftrightarrow \forall k[y