Ludwig Wittgenstein studied with Russell in Cambridge from 1911 to 1913, and wrote the Tractatus while on active service in the Austrian army during the First World War. Large parts of the book are devoted to explaining and correcting errors in the conception of logic to be found in Principia. Wittgenstein did not, as is sometimes suggested, reject the idea of a hierarchy of types, but he did reject the notion that mathematics (and in particular arithmetic) could be based, as in Principia, on classes. For this reason although the account of arithmetic given in the Tractatus is in a sense logicist, it is very different from Russell's.
Keywords: Wittgenstein, Tractatus, language, doctrine of inexpressibility, logicist, proposition, Principia, equations, axiom of infinity