Extending du Bois-Reymond's Infinitesimal and Infinitary Calculus Theory

TL;DR

This paper develops a new infinitesimal and infinitary number system called Gossamer numbers, extending classical calculus with a more general transfer principle and a two-tier system, offering novel insights into infinite and finite sequences.

Contribution

It introduces the Gossamer number system, generalizes the transfer principle beyond Non-Standard Analysis, and establishes a two-tier calculus framework with non-reversible arithmetic.

Findings

Gossamer numbers unify finite and infinite quantities.

A more general transfer principle than Non-Standard Analysis is formulated.

Sequences are partitioned into finite and infinite intervals.

Abstract

The discovery of the infinite integer leads to a partition between finite and infinite numbers. Construction of an infinitesimal and infinitary number system, the Gossamer numbers. Du Bois-Reymond's much-greater-than relations and little-o/big-O defined with the Gossamer number system, and the relations algebra is explored. A comparison of function algebra is developed. A transfer principle more general than Non-Standard-Analysis is developed, hence a two-tier system of calculus is described. Non-reversible arithmetic is proved, and found to be the key to this calculus and other theory. Finally sequences are partitioned between finite and infinite intervals.