On Feb 2, 2:21 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > 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) x=x+1/10^y if (r=2) x=x-1/10^y if (r=3) y=y+1 if (r=4)&(y>1) y=y-1 plot(x,y)
fun infinite-walk() x=0 y=1 repeat random-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 fraction!