Date: Feb 2, 2013 2:38 AM
Author: Graham Cooper
Subject: Re: Outline: A Program to establish the continuity of points in a line

On Feb 2, 2:21 pm, "Ross A. Finlayson" <>
> Decided to start a program.  An outline of my program follows.
> Regards,
> Ross Finlayson
> 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

fun random-walk()
r = rnd(4)
if (r=1)
if (r=2)
if (r=3)
if (r=4)&(y>1)

fun infinite-walk()
until false

Run this for an infinite amount of time and he walks over every point
on the number line!

PROOF: no gaps!

It's an infinite random walk with a twist.
When he moves east or west, he covers 1 unit / 10^y units.


:) ------> :) ------> :)

Here he is moving 1 unit positive at a time.

When y increases - he takes 10 times smaller steps!



y will reach every natural number
and x will be every summation of every possible negative power of 10