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 | 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 | 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 | 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 | 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...