Canonical Set Theory for Classic Mathematics: A Linear Order on Finite Groups and Structures, with a Natural Extension to Infinite Structures

We provide an axiomatic base for the set of natural numbers, that has been proposed as a canonical construction, and use this definition of \(\mathbb N\) to find several results on finite group theory. Every finite group \(G\), is well represented with a natural number \(N_G\); if \(N_G=N_H\) then $H,G$ are in the same isomorphism class. We have a linear order on all finite groups, that is well behaved with respect to cardinality. In fact, if $H,G$ are two finite groups such that $|H|=m<n=|G|$, then \(H<\mathbb Z_n\leq G\). There is also a canonical order for the elements of \(G\) and we can define equivalent objects of \(G\). This allows us to find the automorphisms of \(G\). The Cayley table of \(G\) takes canonical block form, and a minimal set of independent equations that define the group is obtained. We show how to find all groups of order \(n\), and order them. Examples are given using all groups with order smaller than $10$. The canonical block form of the symmetry group \(\Delta_4\) is given and we find its automorphisms. These results are extended to the infinite case. A real number is an infinite set of natural numbers. A real function is a set of real numbers, and a sequence of real functions \(f_1,f_2,\ldots\) is well represented by a set of real numbers, as well. We make brief comments on treating the calculus of real numbers. In general, we represent mathematical objects using the smallest possible data-type. In the last section, mathematical objects are well assigned to tree structures. We conclude with brief comments on type theory and future work on computational and physical aspects of these representations.