What is Russell’s paradox in relation to Gottlob Frege’s logical system?

Russell’s paradox is a fundamental contradiction that exposed a critical flaw in the axiomatic system developed by the German logician Gottlob Frege.21

In 1902, Bertrand Russell wrote to Frege, demonstrating that his system—which relied on an unrestricted comprehension principle—was inconsistent. This principle allowed for the formation of a set based on any condition, including the set of all sets that are not members of themselves ($R=\{x[2][1]|x\notin x\}$). The paradox arises when asking if $R$ is a member of itself: if it is, it must not be; if it is not, it must be.2

This discovery was a devastating blow to Frege’s work, as it undermined the logical foundations he had built for arithmetic. Although Frege attempted to modify his axioms to restore consistency in an appendix to his Basic Laws of Arithmetic, later logicians proved that his adjustments remained insufficient. This crisis effectively ended Frege’s productive career in mathematical logic.1