On Feb 1, 8:37 pm, William Elliot <ma...@panix.com> wrote: > On Fri, 1 Feb 2013, Ross A. Finlayson wrote: > > 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 > > What does that last line all mean? > > > > > > > > > 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
The notion of selecting among discrete items at uniform random is among the most reasonable of assumptions of blind selection among items, that each is as likely as the other. In the finite case, this is well known as fair coin tosses or dice rolls, in Bernoulli and to Poisson. In the infinite, the notion of selecting among the natural integers at uniform random would have particular features of the probability mass function describing the distribution of the natural integers at uniform random. The function from natural integers to points in a line called EF has particular features that would make it at once a cumulative distribution function, over its support space of the natural integers, and in its constant monotonicity, over constant differences, that of a uniform distribution.