How did Russell’s set-theoretic paradoxes affect Frege’s logicist program?

Bertrand Russell’s discovery of Russell’s paradox in 1901 dealt a devastating blow to Gottlob Frege’s logicist program, which aimed to reduce all of mathematics to basic logical laws.123

Frege’s system relied on an unrestricted comprehension principle, which allowed for the formation of a set from any definable condition. Russell demonstrated that this principle permitted the construction of the "set of all sets that are not members of themselves," leading to an impossible contradiction: if the set is a member of itself, it is not; if it is not, it is.143

Upon receiving news of the paradox in 1902, Frege famously remarked that "arithmetic totters". While Frege attempted to modify his axioms to restore consistency, the effort was ultimately unsuccessful. The paradox effectively ended Frege’s productive career in mathematical logic, as he recognized that his foundational goal of grounding arithmetic in logic had been undermined. Consequently, the development of mathematical logic in the 20th century shifted toward alternative approaches, such as Zermelo-Fraenkel set theory and the theory of types developed by Russell and Alfred North Whitehead.3524