00 4 Definability and a Nonlinear Sigma Model

The concept of definability of physical fields in a set-theoretical foundation is introduced. A set theory is selected in which we get mathematics enough to produce a nonlinear sigma model. Quantization of the model requires only a null postulate and is then shown to be necessary and sufficient for definability in the theory. We obtain scale invariance and compactifica-tion of the spatial dimensions effectively. Three interesting examples of the relevance to physics are 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 eigenstates can be mirrored in the theory by a definable set of finite integers. This concept of definability is appropriate because, were there such a physical field whose set of energy distributions among eigenstates was mirrored by 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 observable. Therefore, our task is to find a foundation in which it is possible to specify completely the definable sets of finite integers and to construct the fields mirrored by 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