Decided to start a program. An outline of my program follows.
A Program to establish the continuity of points in a line
The continuum of numbers is a primary feature of mathematics. Logic establishes structures modeling the numbers as abstract things. Most simple concepts of symmetry and conservation establish numerical constructs and identities. Points in a line are built from first and philosophic principles of a logic, and a geometry of points and space. Their continuity is established. Fundamental results of real analysis are established on this line as of the continuum of real numbers. Identities are established for certain fundamental properties of real numbers in a line in the geometry.
An axiomless system of natural deduction Conservation and symmetry in primary objects Categoricity of a general theory Geometry Number theory, analysis, and probability Sets, partitions, types, and categories A natural continuum from first principles The continuum in abstract A continuum of integers The establishment of a space of points from a continuum Drawing of a line in the space of points The polydimensional in space Features of N The infinite in the natural continuum EF as CDF, the natural integers uniformly Features of R Points as polydimensional Results in the polydimensional Continuity in the real numbers Reductio of points in space Topological counterparts of the open and closed Fundamental results of real analysis The complete ordered field in the space of points Fundamental theorems of integral calculus Apologetics Infinitesimals and infinities Rational numbers and exhaustion The continuum as countable Reflection on the drawing of the line as countable Cantor's argument and counterexamples A constructive interpretation of uncountable A retrofit of measure theory Applications Applications in geometry Applications in probability Applications in physics