This book has made visible and described quite a few faults, as if contemporary science, both natural and social, as well as philosophy, were traversed by deep-lying tectonic fractures. I have intentionally adopted a descriptive attitude, my intention being to show these faults without pretending to be able to solve, adjust, or synthesize them. For sure, I have shown one possible direction, namely the idea of placing impredicativity and impredicative systems at the very center of science, and I have called attention to some of the consequences arising from such a radical move. As unpalatable as this move at first appears, it nevertheless shows entirely new avenues for science. Whether science will indeed move in those directions remains to be seen.