The grounding for Continuum

TL;DR

This paper proposes a constructive approach to the mathematical continuum based on adjacency relations, aligning with Brouwer's intuitionistic view and offering a new perspective on the Continuity Theorem.

Contribution

It introduces a formal constructive framework for the continuum using adjacency relations, addressing the lack of a primitive concept in type theory.

Findings

A new constructive definition of the continuum based on adjacency relations.

Discussion of the Brouwer Continuity Theorem without relying on the Axiom of Continuity.

Bridging the gap between abstract set-theoretic notions and constructive intuitionistic perspectives.

Abstract

It is a ubiquitous opinion among mathematicians that a real number is just a point in the line. If this rough definition is not enough, then a mathematician may provide a formal definition of the real numbers in the set theoretic and axiomatic fashion, i.e. via Cauchy sequences or Dedekind cuts, or as the collection of axioms characterizing exactly (up to isomorphism) the set of real numbers as the complete and totally ordered Archimedean field. Actually, the above notions of the real numbers are abstract and do not have a constructive grounding. Definition of Cauchy sequences, and equivalence classes of these sequences explicitly use the actual infinity. The same is for Dedekind cuts, where the set of rational numbers is used as actual infinity. Although there is no direct constructive grounding for these abstract notions, there are so called intuitions on which they are based. A rigorous approach to express these very intuition in a constructive way is proposed. It is based on the concept of the adjacency relation that seems to be a missing primitive concept in type theory. The approach corresponds to the intuitionistic view of Continuum proposed by Brouwer. The famous and controversial Brouwer Continuity Theorem is discussed on the basis of different principle than the Axiom of Continuity.