  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.
