A Relational Axiomatic Framework for the Foundations of Mathematics

TL;DR

This paper introduces a relational axiomatic framework called DGSS that models natural numbers without Peano's axioms, offering new insights into the foundations of mathematics and the nature of zero.

Contribution

It develops a novel relational calculus and a unifying model that derives fundamental mathematical concepts without traditional axioms.

Findings

DGSS models natural numbers without Peano's Axioms

Extensionality and substitution are theorems of DGSS

Provides new perspectives on the concept of zero

Abstract

We propose a Relational Calculus based on the concept of unary relation. In this Relational Calculus different axiomatic systems converge to a model called Dynamic Generative System with Symmetry (DGSS). In DGSS we define the concepts of relational set and function and prove that extensionality and the substitution property of equality are theorems of DGSS. As a first exemplification of DGSS, we construct a model of natural numbers without relying on Peano's Axioms. Eventually, some new clarifications regarding the nature of the number zero are given.