Definability in Physics D.j. Bendaniel

The concept of definability of physical fields in a set-theoretical foundation is introduced and an axiomatic set theory is proposed which provides precisely the tools necessay for a nonlinear sigma model. In this theory quantization of the model derives from a null postulate and becomes equivalent to definability. We also obtain scale invariance and compactification of the spatial dimensions effectively. The applicability of this foundation to quantum gravity is suggested. We look to provide a deep connection between physics and mathematics by requiring that physical fields must be definable in a set-theoretical foundation. The well-known foundation of mathematics is the set theory called Zermelo-Fraenkel (ZF). In ZF, a set U of finite integers is definable if and only if there exists a formula Φ U (n) from which we can unequivocally determine whether a given finite integer n is a member of U or not. That is, when a set of finite integers is not definable, then there will be at least one finite integer for which it is impossible to determine whether it is in the set or not. Other sets are definable in a theory if and only if they can be mirrored by a definable set of finite integers. Most sets of finite integers in ZF are not definable. Furthermore, the set of definable sets of finite integers is itself not definable in ZF. [1] 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 its eigenstates can be mirrored in the 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 is impossible to determine, so the field could not be verifiable. Therefore, our task is to find a foundation in which it is possible to specify completely the definable sets of finite integers and which contains mathematics rich enough to obtain the fields corresponding to these sets. The definable sets of finite integers cannot be specified completely in ZF because there are infinitely many infinite sets whose definability is undecid-able. So we will start with a sub-theory containing no infinite sets of finite 1