BenDaniel, David J.
(2013)
Constructibility in Physics.
In: UNSPECIFIED.
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Abstract
We pursue an approach in which spacetime proves to be relational and its differential properties fulfill the strict requirements of EinsteinWeyl causality. Spacetime is developed from a set theoretical foundation for a constructible mathematics. The foundation proposed is the axioms of ZermeloFrankel (ZF) but without the power set axiom, with the axiom schema of subsets removed from the axioms of regularity and replacement and with an axiom of countable constructibility added. Four arithmetic axioms, excluding induction, are also adjoined; these formulae are contained in ZF and can be added here as axioms. All sets of finite natural numbers in this theory are finite and hence definable. The real numbers are countable, as in other constructible theories. We first show that this approach gives polynomial functions of a real variable. Eigenfunctions governing physical fields can then be effectively obtained. Furthermore, using the integral form of the field equations over a compactified space, we produce a nonlinear sigma model. The Schroedinger equation follows from a proof in the theory of the discreteness of the spacelike and timelike terms of the model. This result suggests that quantum mechanics in this relational spacetime framework can be considered conceptually cumulative with prior physics.
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