Continuum as a Primitive Type: Towards the Intuitionistic Analysis

It is a ubiquitous opinion among mathematicians that a real number is just a point in the (real) line. If this rough definition is not enough, then a mathematician may provide a formal definition of the real numbers in the set theoretical and axiomatic fashion, i.e. via Cauchy sequences or Dedekind cuts, or as the collection of axioms (up to isomorphism) characterizing exactly the set of real number 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 the above 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.