Set Matrix Theory as a Physically Motivated Generalization of Zermelo-Fraenkel Set Theory

TL;DR

This paper introduces set matrix theory (SMT) as a physically motivated extension of Zermelo-Fraenkel set theory (ZF), providing a more suitable foundation for the Elementary Process Theory (EPT) in physics.

Contribution

SMT is proposed as a new set-theoretic foundation that generalizes ZF, enabling formalization of EPT and better aligning with physical applications.

Findings

SMT can construct all sets that ZF can.

SMT is more suitable than ZF for physics applications.

SMT supports formalization of EPT.

Abstract

Recently, the Elementary Process Theory (EPT) has been developed as a set of fundamental principles that might underlie a gravitational repulsion of matter and antimatter. This paper presents set matrix theory (SMT) as the foundation of the mathematical-logical framework in which the EPT has been formalized: Zermelo-Fraenkel set theory (ZF), namely, cannot be used as such. SMT is a generalization of ZF: whereas ZF uses only sets as primitive objects, in the framework of SMT finite matrices with set-valued entries are objects sui generis, with a one-by-one set matrix [x] being identical to the set x. It is proved that every set that can be constructed in ZF can also be constructed in SMT: as a mathematical foundation, SMT is thus not weaker than ZF. In addition, it is shown that SMT is more suitable han ZF for the intended application to physics. The conclusion is that SMT, contrary to ZF, is acceptable as the mathematical-logical foundation of the framework for physics that is determined by the EPT.