 Set theory was beginning to become an essential ingredient of the new “modern” approach to mathematics. But this viewpoint was contested, and its consolidation took a rather long time. Dedekind’s algebraic style only began to find followers in the 1890s; David Hilbert was among them. The soil was better prepared for the modern theories of real functions: Italian, German, French and British mathematicians contributed during the 1880s. And the new foundational views were taken up by Peano and his followers, by Frege to some extent, by Hilbert in the 1890s, and later by Russell.