The 1920s atmosphere, among experts in foundational studies, was one of great insecurity. Most of them - intuitionists excluded - were looking for a symbolic, formal system that might provide a framework for all of mathematics. It had to be possible to reconstruct mathematics on a completely secure basis, to find a system maximally immune to rational doubt.³ But, in carrying out that project, caution was the keyword. Reminders were the by-then legendary paradoxes, which ruined the work of Frege and imperiled that of Cantor, the intuitionists' indictment against modern mathematics, and the proliferation of divergent systems. During the interwar period there was a great level of experimentation in the area of foundations, not infrequently leading to systems that turned out to be contradictory.⁴ Even those who were convinced of the final vindication of the "classical" viewpoint of Cantor and Dedekind, like Hilbert, had to look for very careful ways of proceeding if they wanted to solve satisfactorily all of the problems posed.