The convergence of Fourier series was a major area of research in the 1860s and 1870s. For a "sufficiently regular" periodic function, the result posed no problem, but a strange phenomenon sometimes arose: the series could converge at points where the periodic function was not even continuous.
Antinomies among the ordinals --------------------------------
The German mathematician Georg Cantor is still very young when he tackles this problem. He investigates the sets of discontinuities of such functions, which leads him to develop set theory and, above all, the theory of infinite cardinals. He says that two sets A and B have the same cardinality (whether finite or not) if a bijection can be established between them: each element of A must correspond uniquely to an element of B and, conversely, each element of B must be the image of one and only one element of A. He then shows that the integers and the rational numbers have the same cardinality, and that there are as many points on a line as there are in a plane. But are all infinite cardinals equal? Cantor proves that they are not in an article published in 1874, showing that no bijection exists between the integers and the real numbers. His proof is difficult, and it is not until 1891 that he gives a beautifully clear, historic proof, now known as Cantor's diagonal argument (see Les Ensembles (Sets), Bibliothèque Tangente 61, 2017). At the same time, he establishes that infinitely many transcendental numbers exist—that is, numbers that are not solutions of any polynomial equation with integer coefficients—even though Charles Hermite and Carl Lindemann had faced considerable difficulty in proving the transcendence of e and π in 1873 and 1882.
Cantor seeks to give the various infinite cardinals the status of numbers, thereby introducing ordinal numbers. In particular, he wants to define a total order, as with the integers (which correspond to the finite cardinals). Taking his cue from the latter, he says that the ordinal of a set A is less than (or equal to) that of a set B if there is an injection from A into B; this generalizes what happens in the finite case.
To complete his theory of ordinals, Cantor needs two results. The first states that, given two sets A and B, either there is an injection from A into B or there is one from B into A, ensuring that the order is total. The second, known as the Cantor–Bernstein theorem, states that, given two sets E and F, if there is an injection from E into F and another from F into E, then the two sets are in bijection. The ordinals can therefore be ordered just like finite cardinals.