In the previous chapter, it was shown that the Tractatus accounted for only a limited part of arithmetic at best. When Russell read Wittgenstein's manuscript, he was convinced that this was a gap that needed to be filled. Curiously, it was not the omission of an account of real numbers that he found egregious, but that of a general account of cardinals. Russell was not willing simply to abandon the development of mathematics from the theory of types. He accepted, though, some of Wittgenstein's criticisms of the account he had given in Principia. He took the opportunity presented by the publication of a second edition of Principia to prepare a new Introduction indicating how mathematics could be based on a new theory of types consonant with the parts of Wittgenstein's account that Russell agreed with.
Keywords: Tractatus, Wittgenstein, Russell, Principia Mathematica, atomism, empiricis, hierarchy of propositional functions, mathematical induction, identity