But some of these developments have , in turn, brought about a fresh batch of paradoxical situations The generalization to transfinite numbers and the abstract study of sets have widened and deepened many branches of mathematics, but, at the same time, they have revealed some very disturbing paradoxes which appear to lie in innermost depths of mathematics. Here is where we seem to be today , and it may be that the second half of the twentieth century will witness the resolution of these critical problems.