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

MATHEMATICAL

UNIVERSE

Natural Sets, Mathematical Logic, Set Theory and Mathematical Proofs

The intuitive notion of set and the description of natural sets of numbers **N**, **Z**, **Q**, **R**, **C** and **H** and their elementary properties

Rigorous mathematical expression, based on set theory and mathematical symbolism

The formalization of the essential properties of the sets **N**, **Z**, **Q** and **R**

The basics of naive set theory and the correspondence between set operations and logical syntax

The usual rules of demonstration and their illustration through an elementary theory of the arithmetic-geometric continuum (from natural integers to complex numbers)

**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**.

Natural Sets, Mathematical Logic, Set Theory and Mathematical Proofs

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

**T**his course, the **first volume** of **semester I** of the **first year** of **Mathesis - The Mathematical Universe**, is entirely written by me. It is designed to help anyone who wants to learn mathematics seriously, to **get the best start** in this path. I myself use the approach and advice I offer in this book! The construction of the knowledge and applications presented in this progression are the result of personal work that has provided real results, which have allowed me to go from being a student in retraining to an experienced mathematician.

**O**f course, simply reading this booklet will not turn you into a mathematics 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 analyze demonstrations and do exercises.

**W**hat you will learn and understand in this first course is the quickest and surest way to enter solidly into mathematical knowledge. 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 lay a solid foundation for **your mathematical learning**. You are responsible for your success, but I have done everything I can to guide you step-by-step, without failure, as I have done for many students.

**E-book, 79 pages, .pdf**

17 $

23 lessons, 45 figures, 79 pages, 1 month training

A solid foundation for getting started in Mathematics

Self-study in fail-safe mode

The basics of set theory, from which all objects of modern mathematical science are described

Specific mathematical expression, natural mathematical logic and rigorous use of mathematical symbols

Intuitive properties of natural sets of numbers (natural numbers, integers, rational numbers, real numbers)

The usual proof rules in mathematics, the proof of your first theorems, and an elementary theory of natural sets

23 lessons, 45 figures, 82 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. 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.

23 simple high-level lessons, integrating theory and practice, to give you a solid start to your mathematics learning

Jean Barbet - M A T H E S I S

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