C3: The Canonized Cardinal Continuum

As a viable solution to Cantor’s Continuum Hypothesis (CCH), the C3 treatment of infinitesimals and infinites accounts for hyper-reals consistently with the extension principle, the transfer principle, the real statements, and L’Hopital’s Rule with very specific exceptions. C3 sets a cardinal continuum of infinites and infinitesimals in corresponding reciprocal and algebraic relationships leading to a definition of 1/0 then allowing for the determination of the Indeterminate Forms. These more complete definitions of the role of infinites, infinitesimals, and zero clarify the “ghosts of departed quantities” issues that surround Δx and dx in Differentiation and Integration in Calculus by modifying the Standard Parts Method. The utility of C3 extends beyond the Continuum Hypothesis to Infinitesimal Calculus and Non-Standard Analysis seamlessly.

C3- The Canonized Cardinal Continuum.pdf

Table of Contents

1.0- C3: the Canonized Cardinal Continuum

Introduction

     1.1- An Infinite Series with an Infinitesimal Final Term

     1.2- Degrees of Infinity

     1.3- Infinitesimals as Reciprocals for Degrees of Infinity

     1.4- Roots of Infinites and Infinitesimals

     1.5- Arithmetical Treatments of Infinites and Infinitesimals

     1.6- Range of Value

     1.7- Perambulation

     1.8- Reciprocating Perambulations

     1.9- Arithmetical Treatments of Perambulations

     1.10- Subambulation

     1.11- Ambulation

     1.12- Central Core Values and Super Order

     1.13- Defining the Reciprocal of Zero

     1.14- Un-ordinals and the Non-set

     1.15- Counting the Continuum using Powersets

     1.16- Limits

2.0- C3 and the Conventional Approach

     2.1- Antithetical Proof

     2.2- Compatibility of C3 with Non-Standard Analysis

     2.3- Standard Parts Method in Differentiation

     2.4- Standard Parts Method in Integration

     2.5- Failure of the Standard Parts Method

     2.6- Determining the Indeterminate Forms

     2.7- C3 and Differentiation

     2.8- C3 and Integration

Bibliography