Another New Foundation: A Theory with Combined Concepts from Set Theory, Type Theory and Le\'{s}niewski's Mereology

TL;DR

This paper proposes a new foundational theory for mathematics combining set theory, type theory, and mereology, aiming to serve as an alternative to ZFC with its own axioms and realizations of mathematical structures.

Contribution

It introduces a novel foundational framework integrating multiple logical concepts and discusses its axioms, realizations, and comparison with traditional set theory ZFC.

Findings

The theory successfully models natural numbers, integers, rationals, reals, and complex numbers.

It offers an alternative foundation to ZFC with distinct axioms.

The paper discusses the mathematical limitations of the new theory.

Abstract

This paper introduces a new theory which encompasses concepts and ideas from set theory, type theory, and Le\'{s}niewski's mereology and describes its possibility as an alternative foundation for mathematics. In the introduction section I will introduce motives for development of the theory and some remarks on the methods of presentation. Axioms of the theory and their philosophical background and justification on the basis of intuitive view are discussed next. Discussed after are realizations of mathematical concepts such as $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, etc. Then this paper concludes with comparisons between the theory and ZFC and its mathematical limitations.