Why did Russell’s paradox challenge Frege’s logicist program?

Gottlob Frege’s logicist program aimed to reduce all of mathematics to basic logical laws. A cornerstone of his system was the comprehension principle, which allowed for the formation of a set for any condition expressible by a formula.123

Russell’s paradox challenged this program by demonstrating that Frege’s unrestricted comprehension principle was inconsistent. Specifically, Bertrand Russell showed that one could define the set of all sets that are not members of themselves. This leads to a logical contradiction: if the set is a member of itself, it must not be; if it is not, it must be.24

Because Frege’s system relied on this principle to define mathematical concepts, the paradox revealed a fundamental flaw in his axiomatic foundation. This forced a reassessment of how to construct sets and define mathematical objects without falling into the "vicious circle" of self-referential definitions. Consequently, the paradox necessitated the development of more restrictive systems, such as the theory of types or Zermelo-Fraenkel set theory, to restore consistency to the foundations of mathematics.234