L'infini au cours des siècles

The fascinating history of the concept of infinity is indissolubly associated with the development of religions, philosophies, mathematics and natural sciences. At the time of the first Greek philosophers, the infinity, or áperion, was a negative concept to be avoided in all rigorous statements. Later, Aristotle partially accepted the idea of infinity by distinguishing potential infinity from actual infinity. While the former was acceptable, being represented by a growing sequence with no last element, the latter was regarded as impossible, being idealized with a complete infinite sequence consisting of infinitely many elements. These earlier concepts are the basis of all the following developments that led thinkers like Euclid, Archimedes, Galilei, Cauchy, Dedekind, Cantor, Gödel and many others to clarify the concept of infinity and to reach its modern formulation. We propose here a brief outline of the history of the concept of infinity, with particular emphasis on its mathematical definition and implications.