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

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 $2^{X}$.