 Though Frege’s system was subject to Russell’s Paradox, subsequent examination of his system shows how much of the development of arithmetic is possible independently of the paradoxical elements of the system. In particular, recent interest in Frege’s system has led to the isolation of what is called “Frege’s Theorem” as possible in a consistent fragment of Frege’s original system, and from it the goal of deriving arithmetic, as formalized in Peano’s Postulates. See the entry  Frege’s theorem and foundations for arithmetic,  which presents this aspect of Frege’s system in contemporary notation.