**M A T H E S I S**

APPLICATIONS

AND NUMERATION

From Finite Sets to Mathematical Infinity

The notions of Cartesian product, relation and application, and all concepts associated with applications or functions (image, antecedent, reverse image, restriction, composition...)

The rigorous mathematical formalisation of the notion of "number of elements" of a set, based on the theory of bijective applications and the concept of equipotence

The numeration of finite sets, and the denumeration of finite sets obtained by elementary set constructions, binomial coefficients and permutations

The precise definition of infinite sets, and two essential characterisations of mathematical infinity, from the set **N** of natural numbers and intrinsically

Preparation for the axiomatic study of arithmetic and natural sets, based on the properties of the successor function

**M**y name is Jean Barbet, I am an independent mathematician and I have been teaching mathematics for more than ten years. I hold degrees in experimental sciences (Master's degree in Biology and postgraduate degree in Environmental Sciences) and a PhD in **Pure Mathematics** (Lyon 1 University, 2010). I

**A**fter an experience of techno-scientific research in Ecophysiology, I reconverted to Mathematics. Enrolled in the 3rd year of a Bachelor's degree at the University, I experienced the material limits of university teaching (restriction of the volume of contents, concentration on algebra and analysis, multiple approaches...). I filled my gaps and completed my learning by undertaking an **integration of mathematical knowledge**. By adding substantial but fundamental elements - especially in number theory, geometry and mathematical logic - I have developed a "system of higher mathematics", called **Mathesis**.

**I** wish to transmit this system as a written corpus and an online curriculum to make the **core of essential mathmatics** accessible to all. The corpus is conceived as a complete and self-sufficient body of knowledge, corresponding to a Bachelor's degree in mathematics (Bac +3) of the highest level, plus substantial supplements. The aim of Mathesis is to enable anyone to learn the essentials of higher mathematics on their own.

**Y**ou can find more information on the corpus and a detailed description of the approach and programme at **Mathesis: integrating mathematical knowledge** and all related publications at **Mathesis: e-Books**.

From Finite Sets to Mathematical Infinity

M A T H E S I S - The Mathematical Universe - Year 1, Semester I, Course n°2

This **second course** or volume of **Semester I** of the **first year** of Mathesis - The Mathematical Universe, is entirely my own. It is designed to help you quickly and solidly grasp the essential and structuring concepts of **mathematical finiteness and infinity**.

It is an opportunity to complete the **basics of naive set theory** acquired in the first course, by the fundamental notions of Cartesian product, relation and application, which allow to develop the **rigorous theory of functions**, from which we can really talk about the "**number of elements**" of a set (or rather, about the notion of equipotence, which mathematically compares the numbers of elements).

It is also the occasion to approach the bases of "combinatorics" or **theory of finite sets**, through the numeration of these sets (i.e. the attribution of a cardinal or number of elements) and the **enumeration** of the finite sets which one can build starting from other finite sets; it is notably a question of binomial coefficients and of permutations.

The **theory of infinite sets**, defined from finite sets, is based on the same principles, and gives rise to their intrinsic characterisation by the primitive concepts of set theory, which is already a significant achievement in mathematical learning. Between the finite and the infinite, we approach the essential study of the set **N** of natural numbers, through its **successor function**, on which rests the axiomatisation of arithmetic that we will tackle in the following course, and the study of other natural mathematical sets, which will be tackled in the rest of the semester.

Of course, merely reading this booklet will not turn you into a mathematical expert overnight. You will have to follow the work advice I give you at the beginning of the course, and above all **work with regularity and perseverance**, without ever getting discouraged. I tell you more in the book, but you will have to analyse demonstrations and do exercises.

What you will learn and understand in this **second course** is a safe and accessible path to the knowledge of mathematical finiteness and infinity; the content of the first course is, however, considered as a prerequisite, especially what concerns naive set theory and elementary logic. Mathematics is first and foremost a science, so you will find theory; but you understand by doing, and you will often have to solve exercises and problems to implement the course, which I will suggest in most lessons. You must be aware of the effort involved, and that reading this book like a novel will not be enough to tame mathematical infinity. You are responsible for your own success, but I have done my best to guide you step by step, without failure, as I have done for many students.

**E-book, 66 pages, .pdf **

17 $

18 lessons, 23 figures, 66 pages, 1 month training

A complete course on finite set theory and mathematical infinity

Self-study in fail-safe mode

The complements of set theory (Cartesian products, relations and applications) essential to all higher mathematics

The elementary theory of mathematical functions: injectivity, surjectivity, bijectivity and number of elements or "cardinality"

The basics of combinatorics or finite set theory, through numbering and enumeration

A fundamental study of mathematical infinity, culminating in an intrinsic characterisation of infinite sets, and preparing the axiomatic study of arithmetic

18 lessons, 23 figures, 66 pages, numerous examples and exercises at an accessible level to illustrate and complement the theory

How can I access the e-book?

Once you have purchased the e-book, you will receive an e-mail with access to a space dedicated to the students of the course. There you will find the e-book in .pdf format, which you can then download. You will be able to start your mathematical curriculum immediately.

Do I need previous knowledge of mathematics?

The principle of Mathesis, the programme of which this book is the first step, is to start from scratch by laying a sound and rigorous foundation. The prerequisites for any course are some contents of the preceding courses. If you start from the first one, middle and high school mathematics might be a useful intuitive foundation, though we reintroduce them as we go along. You will have no difficulty in finding on your own the external background knowledge you may need from time to time.

I can't find my e-book anymore. Can I download it again?

Of course. You will keep your access to your member area, and you will be able to download your e-book for your personal use as often as you need it. There will also be revisions and you will be notified by e-mail of new versions that are published.

18 advanced lessons, integrating theory and practice, for a rigorous understanding of mathematical finiteness and infinity

Jean Barbet - M A T H E S I S

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