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## The higher axioms of natural 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...

## Natural Set Theory: An Ultimate Foundation for Mathematics

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...

## The axiomatic construction of natural arithmetic

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...

## Russell’s paradox and the emergence of class 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...

## An infinity of prime numbers : Euclid’s theorem

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...

## What is a rational number? Quotients of numbers and sets

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...