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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.

\medskip
\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}
\end{center}
%\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, $y=0 \leftrightarrow \forall k[y<l/k]$,
where $k$ is a finite integer.  The reciprocal of an infinitesimal is
``infinite''.  A member of Q$^*$ that is not an infinitesimal and not
infinite is ``finite''.  The constructibility axiom well-orders the
power set of $\omega^*$, creating a metric space composed of the
subsets of $\omega^*$.  These subsets represent the binimals making up
a real line R$^*$.

\medskip
\noindent
An {\it equality-preserving} bijective map $\Phi(x,u)$ between
intervals X and U of R$^*$ in which $x \in X$ and $u \in U$ such that
$\forall x_1,x_2,u_1,u_2[\phi(x_1,u_2)\wedge \phi(x_2,u_2)\to (x_1, -
x_2 =0\leftrightarrow u_1 - u_2 =0)]$ creates pieces that are biunique
and homeomorphic.  It is clear that $U$ vanishes if and only if $X$
vanishes.

\medskip
\noindent We can now define ``functions of a real variable in T''.
$u(x)$ is a function of a real variable in T only if it is a constant
or a sequence in $x$ of continuously connected biunique pieces such
that the derivative of $u$ with respect to $x$ is also a function of a
real variable in T. These functions are thus of bounded variation.  If
some derivative is a constant, they are polynomials.  If no derivative
is a constant, these functions do not exist in T.  They can, however,
be approached arbitrarily closely by some linear combination of
polynomials of very high degree given by many iterations of the
following variational form for the Sturm-Liouville problem, so that
these polynomials are effectively eigenfunctions:

\noindent $\int\limits_a^b[p\left(\frac{du}{dx}\right)^2
-qu^2]dx\equiv  \lambda\int\limits_a^b ru^2dx: \lambda$ is
locally minimum for $\int\limits_a^b ru^2dx$ constant;

\noindent where a $\not = b, u\left(\frac{du}{dx}\right)\equiv 0$ at
a and b; $p,q,r$ are functions of the real variable $x$


\medskip
\noindent Let $u_1$ and $u_2$ be functions of the real variables $x_1$
and $x_2$ respectively.  Let us consider a one-dimensional string,
where $x_1$ is space and $x_2$ is time.  We can write Hamilton's
Principle using a null identity, since:

\noindent $\int\left[\left\lgroup\frac{\partial u_1u_2}{\partial
x_1}\right\rgroup^2-\left\lgroup\frac{\partial u_1u_2}{\partial
x_2}\right\rgroup^2\right]dx_1dx_2\equiv 0\to \delta
\int\left[\left\lgroup\frac{\partial u_1u_2}{\partial
x_1}\right\rgroup^2-\left\lgroup\frac{\partial u_1u_2}{\partial
x_2}\right\rgroup^2\right]dx_1dx_2=0$

\medskip
\noindent A generalization to finitely many space-like and time-like
dimensions can now be obtained.  Let $u_{\ell mi}(x_i)$ and $u_{\ell
mj}(x_j)$ be eigenfunctions with non-negative eigenvalues
$\lambda_{\ell mi}$ and $\lambda_{\ell mj}$ respectively. We assert a
``field'' is a sum of eigenstates: $\underline{\Psi}_m
=\sum\limits_\ell \Psi_{\ell m}\underline{i_\ell}, \Psi_{\ell m} =
\prod\limits_i u_{\ell mi} \prod\limits_j u_{\ell m j}$, subject to
the postulate that for every eigenstates $m$ the value of the integral
of the Lagrange density over all space-time is {\it identically} null.
Let $dsd\tau =\prod\limits_i r_i dx_i \prod\limits_j r_j dx_j$:

\noindent $\sum\limits_{\ell}\int \left\{\sum\limits_i \frac{1}{r_i}
\left[P_{\ell m i}\left\lgroup\frac{\partial\Psi_{\ell m}}{\partial
x_i}\right\rgroup^2 -Q_{\ell mi}\Psi_{\ell m}^2\right]- \sum\limits_j
\frac{1}{r_j}\left[P_{\ell m j}\left\lgroup\frac{\partial\Psi_{\ell
m}}{\partial x_j}\right\rgroup^2 -Q_{\ell mj}\Psi_{\ell 
m}^2\right]\right\}dsd\tau
\equiv 0 for ~ all~ m$.

\medskip

\noindent In this integral expression the $P$ and $Q$ can be functions
of any of the $x_i$ and $x_j$, thus of any $\Psi_{\ell m}$ as well.
This is a {\it nonlinear sigma model}.  The $\Psi_{\ell m}$ can be
determined by an algorithm with coordinated iterations that are
constrained by the indicial expression $\sum\limits_{\ell i}
\lambda_{\ell mi}- \sum\limits_{\ell j}
\lambda_{\ell mj}\equiv 0$ {\it for all m}.

\medskip

\noindent The following is a proof of quantization using the
mathematics of T plus the null postulate.  Let

\noindent $\sum\limits_{\ell mi}\int \left\{\frac{1}{r_i}\left[P_{\ell
mi}\left\lgroup\frac{\partial\Psi_{\ell m}}{\partial
x_i}\right\rgroup^2 -Q_{\ell mi}\Psi_{\ell m}^2\right]\right\}dsd\tau$
\\ and $\sum\limits_{\ell mj}\int \left\{\frac{1}{r_j}\left[P_{\ell
mj}\left\lgroup\frac{\partial\Psi_{\ell m}}{\partial
x_j}\right\rgroup^2 -Q_{\ell mj}\Psi_{\ell m}^2\right]\right\}dsd\tau$
both be represented by $\alpha$, since they are identical:

\medskip

I.~~~~~$\alpha$ is positive and must be closed to addition and to
the\\ \indent~~~~~~ absolute value of subtraction; In T we must have 
that
$\alpha$ is an\\\indent~~~~~~ integer times a constant which is
infinitesimal or finite.\\ \indent II.~~~There is either no field, in
which case $\alpha \equiv 0$, or otherwise in T\\ \indent~~~~~~ the
field is finite in which case $\alpha \neq 0$); thus $\alpha = 0
\leftrightarrow \alpha \equiv 0$.\\ \indent III.~~Therefore $\alpha
\equiv nI$, where n is an integer and $\it I$ is a finite constant
\indent~~~~~~~such that $\alpha = 0 \leftrightarrow n \equiv 0$

\medskip
\noindent Quantization is thus obtained within this theory utilizing
both the relational nature of space-time in T and a physical null
postulate.  The Schr$\ddot o$dinger equation can now be derived for
finitely many space variables and one time variable.  Thence we can
get the eigenstates and the energy levels associated with each
state.  When there are finitely many space-like dimensions and one
time-like dimension, we can obtain the Schr$\ddot o$dinger equation
from the nonlinear sigma model: Let $\ell$ = 1,2 and suppress $m$.  We
can now introduce $\Psi = A\prod\limits_{i} u_i
(x_i)[u_1(\tau)+ \iota u_2(\tau)]$, where $\iota$= $\sqrt-1$, normalizing
$[u_1^2(\tau)+u_2^2(\tau)]\int \prod\limits_{i}u_i^2(x_i)ds\equiv 1$.
Then we see that $\frac{du_1}{d\tau}= -u_2$ and $\frac{du_2}{d\tau} =
u_1$ or $ \frac{du_1}{d\tau} = u_2$ and $\frac{du_2}{d\tau} = -u_1$.
In either case, for each and every {\it irreducible} binunique
time-eigenfunction piece the least non-zero value of $\alpha$ is the
finite constant $\it I$.  Thus $A^2\int
\int_0^{\pi/2}\left[\left\lgroup\frac{du_1}{d\tau}\right\rgroup^2 +
\left\lgroup\frac{du_2}{d\tau}\right\rgroup^2\right]\prod\limits_{i}
u_i^2 (x_i) ds d\tau \equiv A^2\pi/2\equiv I$.  Substituting the
Planck constant $h$ for 4$\it I$ and letting $\tau$ be $\omega t$, the
integrand becomes the well-known time-part of the Lagrange density for
the Schr$\ddot o$dinger equation, $(h/4\pi
\iota)\left[\Psi^*\left\lgroup\frac{\partial\Psi}{\partial t}\right\rgroup
- \left\lgroup\frac{\partial\Psi^*}{\partial
t}\right\rgroup\Psi\right]$.  It immediately follows that the energy
in the $m^{th}$ eigenstate, obtained from this time-part of the Lagrange
density, exists only in quanta of $h\omega_m/2\pi$.  The sum of
energies in all of the eigenstates $E_t$ is thus $\Sigma
n_mh\omega_m/2\pi$ where $n_m$ is the number of quanta in the $m^{th}$
eigenstate.  For any finite energy, every ordered set of finite
integers ${n_m}$ corresponding to the number of quanta residing in
each of the eigenstates, using the fundamental theorem of arithmetic,
maps bi-uniquely with $\prod\limits_m \huge P_m^{n_m}$, where $\huge
P_m$ is the $m^{th}$ prime starting with 2.  The set of these finite
integers must always be definable in T.  Furthermore, conversely, for
classical physics a definable field does not exist in T.  Given finite
$E_t$ and $\omega_m$, then, if h were infinitesimal, some $n_m$ would
have to be infinite and thus the set of all distributions of
energy among the eigenstates could not be mirrored by any set (in T)
of finite integers.  Accordingly, {\bf definability in T is equivalent
to quantization.}

\medskip

\noindent In summary, \\
\noindent $\Box$ Quantization is derivable from a null postulate in 
the set-theoretical\\ \indent foundation T.\\
\noindent $\Box$ Schr$\ddot o$dinger equation and, more generally, a 
nonlinear sigma model,\\ \indent are obtained.\\
\noindent $\Box$ There are inherently no singularities in these 
physical fields.\\
\noindent $\Box$ Space-time is relational, giving a possible 
foundation for quantum gravity.

\noindent In addition, though we do not have the opportunity to 
discuss these points in any detail, it should be more or less obvious 
that\\
\noindent$\Box$ By similar reasoning, definability in T can be shown 
equivalent to\\ \indent compactification of all the spatial dimensions 
effectively.\\
\noindent $\Box$ Dyson's problem regarding divergence of perturbation 
series in QED goes\\ \indent away.\\
\noindent $\Box$ Wigner's metaphysical question regarding the apparent 
unreasonable\\ \indent effectiveness of mathematics in physics is 
directly answered.

\bigskip
\medskip

\begin{thebibliography}{99}

\medskip

\bibitem{1} Tarski, A., Mostowski, A., Robinson, R., {\it Undecidable 
Theories. }  North Holland, Amsterdam, 1953.

\medskip
\bibitem{2} G$\ddot o$del, K., The consistency of the axiom of choice 
and of the generalized continuum hypothesis.  {\it Annals of Math. 
Studies}, 1940.

\end{thebibliography}

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