$\sigma$-Set Theory: Introduction to the concepts of $\sigma$-antielement, $\sigma$-antiset and Integer Space

TL;DR

This paper introduces $\sigma$-Set Theory, a new framework extending classical set theories, featuring $\sigma$-antielements, $\sigma$-antisets, and a novel algebraic structure called Integer Space, which generalizes the power set algebra.

Contribution

It develops a new axiom system for $\sigma$-Set Theory, introducing $\sigma$-antielements and $\sigma$-antisets, and constructs the Integer Space algebraic structure as a completion of the power set.

Findings

Existence of $\sigma$-antielements and $\sigma$-antisets

Construction of Integer Space as a non-associative algebraic structure

Integer Space generalizes the algebra of power sets

Abstract

In this paper we develop a theory called $\sigma$-Set Theory, in which we present an axiom system developed from the study of Set Theories of Zermelo-Fraenkel, Neumann-Bernays-Godel and Morse-Kelley. In $\sigma$-Set Theory, we present the proper existence of objects called $\sigma$-antielement, $\sigma$-antiset, natural numbers, antinatural numbers and generated $\sigma$-set by two $\sigma$-sets, from which we obtain, among other things, a commutative non-associative algebraic structure called Integer Space $3^{X}$, which corresponds to the algebraic completion of $2^{X}$.