by Jean Barbet | Dec 10, 2024 | Logic, Number Theory, Set Theory
We explore the foundation of natural arithmetic starting from Peano’s axioms within set theory, revealing an innovative approach to conceptualizing natural numbers. We question the traditional use of ordinals and propose an alternative formulation of the axiom...
by Jean Barbet | Jun 8, 2024 | Logic, Set Theory
Based on the notions of object and class derived from natural logic, we have redefined the concept of set in an intuitive way, thus establishing a natural set theory without resorting to formal logic. This approach is based on six axioms that form the basis of this...
by Jean Barbet | May 24, 2024 | Logic, Set Theory
The revolution in mathematics is that of set theory, which responds both to the problem of a universal and rigorous conceptual language and to that of a single foundation for all mathematical disciplines. Although set theory was Cantor’s original work, the...
by Jean Barbet | Sep 26, 2023 | Logic, Non classé, Number Theory, Set Theory
Natural numbers have two faces: on one hand, they can be seen as sequences or “enumerations”—what we call ordinal numbers. On the other hand, they are perceived as “quantities,” which leads us to cardinal numbers. While this distinction is not...
by Jean Barbet | Jul 9, 2023 | Logic, Number Theory, Set Theory
Natural arithmetic is the science of natural numbers: it is based on addition, multiplication, natural order and divisibility. Now, all these operations and relations are defined on the basis of the single successor function, whose properties are brought together in...
by Jean Barbet | Jun 22, 2021 | Algebra, Geometry, Non classé
The visual intuition through which we represent the Euclidean plane suggests that we can orient it according to a direction of rotation. This intuition reflects a rigorous mathematical definition of the orientation of the plane, which involves choosing a basis and,...
by Jean Barbet | May 29, 2021 | Logic, Set Theory
Russell’s paradox or antinomy is a very simple paradox in naive set theory, which arises when one tries to define a “set of all sets”. Its resolution relies on the introduction of the notion of class and the distinction of sets among classes. Thanks...
by Jean Barbet | May 23, 2021 | Algebra, Geometry
The linear transformations of the Euclidean plane are the invertible linear applications, i.e. of non-zero determinant. They allow us to move from one basis of the plane to another, and the orthogonal transformations, i.e. the vectorial isometries, exchange the...
by Jean Barbet | May 8, 2021 | Algebra, Geometry, Non classé
The representation of the Euclidean plane as the Cartesian product \(\mathbb R^2\) allows us to decompose any vector of the plane into two coordinates, its abscissa and its ordinate. This decomposition is linked to a particular and natural “representation...
by Jean Barbet | Mar 25, 2021 | Algebra, Geometry, Non classé, Number Theory
Descartes’ analytical method, which allows the Euclidean plane to be represented as the Cartesian product $ \mathbb{R}^2 $ through the theory of real numbers, also makes it possible to represent Euclidean space as the Cartesian product $ \mathbb{R}^3 =...
by Jean Barbet | Mar 20, 2021 | Algebra, Geometry, Non classé, Number Theory
The complex multiplication naturally extends to a multiplication in four dimensions, which defines on the space $ \mathbb{R}^4 $ the structure of the algebra $ \mathbb{H} $ of Hamilton’s quaternions. This multiplication can be interpreted geometrically using the...
by Jean Barbet | Mar 12, 2021 | Algebra, Non classé, Number Theory
Gaussian integers are complex numbers with integer coordinates. Thanks to their norm, a kind of integer measure of their size, we can describe some of their arithmetic properties. In particular, we can determine which are the usual prime numbers that...
by Jean Barbet | Feb 20, 2021 | Functions, Number Theory
Introduction When we introduced the circular exponential, the trigonometric functions cosine and sine were defined as its real part and imaginary part. From this, we derived the analytical expressions: \(\cos x=\sum_{n=0}^{+\infty} (-1)^n\dfrac{x^{2n}}{(2n)!}\) and...
by Jean Barbet | Feb 13, 2021 | Algebra, Geometry, Non classé
Introduction In Vector angles: geometric intuition and algebraic definition, we defined and described the group of Euclidean plane vector angles algebraically, using an equivalence relation on unit vectors. Just as we can measure lengths, we learn at primary school...
by Jean Barbet | Feb 6, 2021 | Algebra, Geometry, Non classé
Vector angles are the usual oriented angles of Euclidean plane geometry. Thanks to the resources of naive set theory, they can be defined purely algebraically using an equivalence relation and the vectorial rotations of the plane. The operation of composing rotations...
by Jean Barbet | Jan 26, 2021 | Algebra, Geometry
by Jean Barbet | Jan 9, 2021 | Analysis, Functions, Non classé
From the complex exponential function, we can define a “circular exponential” function, which “wraps” the real line around the trigonometric circle, and makes it possible to rigorously define the cosine and sine trigonometric functions, which...
by Jean Barbet | Dec 29, 2020 | Analysis, Functions, Non classé
Some functions that can be differentiated indefinitely can be described ‘around each point’ as the sum of an power series. These are analytic functions, real or complex, the typical example being the exponential function, which can be extended to the whole complex...
by Jean Barbet | Dec 16, 2020 | Non classé, Number Theory, Set Theory
The prime natural numbers are those which have no divisors other than 1 and themselves. They exist in infinite number by Euclid’s theorem, which is not difficult to prove. 1.Prime numbers 1.1.Divisors and primes A prime number is a non-zero natural number (see...
by Jean Barbet | Dec 6, 2020 | Analysis, Functions
The relations between the properties of monotonicity, continuity and derivation of a function of one real variable allow us to formally derivate the inverse bijection of an injective and derivable function. The most representative example is perhaps that of the...
by Jean Barbet | Nov 20, 2020 | Non classé, Number Theory, Set Theory
1.The intuition of rational numbers Rational numbers, i.e. “fractional” numbers, such as \(-\frac 1 2, \frac{27}{4}, \frac{312}{-6783},\ldots\), form an intuitive set which we note \(\mathbb Q\). It is an extension of the set \(\mathbb Z\) of integers (see...
by Jean Barbet | Nov 10, 2020 | Number Theory, Set Theory
Integers are an extension of the natural numbers where the existence of subtraction provides a more appropriate framework for certain questions of arithmetic. They can be described axiomatically, but can also be constructed from the set of natural numbers and some...
by Jean Barbet | Oct 25, 2020 | Geometry, Non classé, Trigonometry
The trigonometric circle allows us to define the cosine, sine and tangent of an oriented angle, and to give an interpretation through Thales’ and Pythagoras’ theorems. Introduction: trigonometry and functions Trigonometry is the study of the relationships...
by Jean Barbet | Oct 4, 2020 | Algebra, Geometry
The scalar or dot product of two vectors in real space is a real number that takes into account the direction, sense and magnitude of both vectors. 1.The natural scalar product in the Euclidean plane 1.1.From the distance between two points to the scalar product In...
by Jean Barbet | Sep 16, 2020 | Algebra, Functions, Non classé
Polynomials with one variable are mathematical representations of the expressions used in polynomial equations. They allow algebraic methods to be applied to solving these equations. 1. Equations are “linguistic” objects 1.1 Polynomial equations and number...
by Jean Barbet | Aug 20, 2020 | Algebra, Non classé
There are various ways of defining complex numbers. The most direct way is to look at them as points or vectors of the Euclidean plane. Addition and multiplication are then defined using the coordinates. 1. The set \(\mathbb C\) of complex numbers 1.1. A complex...
by Jean Barbet | Jul 10, 2020 | Functions, Set Theory
A finite set is a set that can be counted using the natural numbers \(1,\ldots,n\) for a certain natural number \(n\). But what is counting ? And then, what is an infinite set? 1.Comparing sets : the notion of bijection The notions of finite set and infinite set, and...
by Jean Barbet | Jul 6, 2020 | Functions, Geometry
The definition of a circle is simple: it is a set of points located at the same distance from a given point. This distance is called the radius and this point is called the centre of the circle. The circle with centre \((-1,-3/2)\) and radius \(\sqrt 6\) 1. Circles as...
by Jean Barbet | Jul 1, 2020 | Algebra, Geometry
From Descartes’ analytic approach, which consists in introducing coordinates to represent the points of the plane, and from Cauchy’s construction of the real numbers, we can give a modern representation of the euclidean plane from which we recover...
by Jean Barbet | Jun 25, 2020 | Functions, Geometry
The derivative of a function is its instantaneous variation, i.e. the slope of the tangent to the graphical representation of the function at that point. 1. General idea: an instantaneous variation We place ourselves here in the framework of functions of a real...
by Jean Barbet | Jun 24, 2020 | Set Theory
Naive set theory or “potato science” is the natural (and understandable!) foundation of mathematical science. “I know what time is. If you ask me, I don’t know anymore.” Augustine. This insightful quote from Augustine emphasises that...
by Jean Barbet | Jun 22, 2020 | Number Theory, Set Theory
Mathematical science does not seek to define the notion of a natural number, but to understand the set of natural numbers. “Natural numbers have been made by God, everything else is the work of men”. Leopold Kronecker 1.We don’t define the natural...
by Jean Barbet | Jun 20, 2020 | Number Theory, Set Theory
The real numbers are all the “quantities” that we can order, and we can “construct” them in various ways thanks to set theory. “Numbers govern the world.” Pythagoras The real numbers idealise all the “points” of the...
