Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

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

Version 1 : Received: 19 July 2020 / Approved: 19 July 2020 / Online: 19 July 2020 (15:28:45 CEST)
Version 2 : Received: 4 August 2020 / Approved: 5 August 2020 / Online: 5 August 2020 (05:43:50 CEST)
Version 3 : Received: 18 September 2021 / Approved: 20 September 2021 / Online: 20 September 2021 (15:11:14 CEST)
Version 4 : Received: 13 August 2022 / Approved: 15 August 2022 / Online: 15 August 2022 (04:55:03 CEST)

How to cite: Ramirez, J. Canonical Set Theory for Classic Mathematics: A Linear Order on Finite Groups and Structures, with a Natural Extension to Infinite Structures. Preprints 2020, 2020070415 (doi: 10.20944/preprints202007.0415.v2). Ramirez, J. Canonical Set Theory for Classic Mathematics: A Linear Order on Finite Groups and Structures, with a Natural Extension to Infinite Structures. Preprints 2020, 2020070415 (doi: 10.20944/preprints202007.0415.v2).

## Abstract

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.

## Keywords

Structuralism; Set Theory; Type Theory; Arithmetic Model; Data Type; Tree; Group

## Subject

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

Comment 1
Commenter: Juan Ramirez
Commenter's Conflict of Interests: Author
Comment: Several statements in Section 1 and Section 6 have been made clearer.
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