A solution for the differences in the continuity of continuum among mathematicians

TL;DR

This paper proposes a mathematical model that constructs a set in contact everywhere, aiming to resolve longstanding differences among mathematicians regarding the concept of continuum's continuity.

Contribution

It introduces a novel set construction using Dedekind's cut and weakened order axioms to unify the concept of continuum's continuity.

Findings

Constructed a set in contact everywhere

Proved the set can eliminate differences in the continuity of continuum

Uses a modified order axiom to achieve contact everywhere

Abstract

There are the longstanding differences in the continuity of continuum among mathematicians. Starting from studies on a mathematical model of contact, we construct a set that is in contact everywhere by using the original idea of Dedekind's cut and weakening Order axioms to violate Order axiom 1. It is proved that the existence of the set constructed can eliminate the differences in the continuity.