Frege, Friedrich Ludwig Gottlob (1848–1925)
Friedrich Frege was a German mathematician and philosopher who, inspired by the earlier work of Leibniz, virtually founded the modern discipline of mathematical logic. In Die Grundlagen der Arithmetik (1884), he used set theory to define the cardinal number of a given class as the class of all classes that are similar (i.e. can be placed in a one-to-one correspondence) to the given class. In Grundgesetze der Arithmetik (2 vols., 1893 and 1903), Frege began attempting to build up mathematics from arithmetic and symbolic logic on a rigorous and contradiction-free basis. When the second volume was in the process of being printed, Bernard Russell pointed out a paradox in Frege's work. The paradox, which became known as Russell's paradox, stems from the question: "Is the class of all classes that are not members of itself a member of itself or not?" The question leads to a contradiction and cannot be resolved. Frege was thus forced to admit that the foundation of his reasoning was worthless. As he stated at the end of his work, "A scientist can hardly encounter anything more undesirable than to have the foundation collapse just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was almost through the press."