On the path of its development, mathematics periodically experiences turning points, and these crises each time force thinkers to open more and more new horizons. The desire for an ever-increasing degree of abstraction and increasing the rigor of mathematical reasoning inevitably led to reflections on the foundations of mathematics itself and the logical laws on which it relies. However, it is precisely in logic, as has been known since the time of Zeno of Elea, that paradoxes lurk—statements that are insoluble at first (and even at second) glance, which, on the one hand, threaten to destroy many harmonious theories, and on the other, give impetus to their new understanding. The names of David Hilbert, Bertrand Russell, Kurt Gödel, Alan Turing are associated precisely with the birth of completely new points of view on seemingly well-studied phenomena. So let's repeat the amazing path that these scientists took while building a new foundation for mathematics.