Wittgenstein's response to Russell's paradox

Question

Can someone explain Wittgenstein's response to Russell's paradox in the Tractatus? Is it possible to cast the response as a mathematical proof? All explanations I have found so far mix logical and philosophical terms, so I am uncertain what is actually being expressed.

Answer 1

W's Tractatus is not a mathematical work.

Having said that, the Tractatus relies heavily on the logic of Principia Mathematica and the Theory of types.

In a nutshell, W's approach is [see 3.333] :

A function cannot be its own argument, because the functional sign already contains the prototype of its own argument and it cannot contain itself.

If, for example, we suppose that the function $F(fx)$ could be its own argument, then there would be a proposition “$F(F(fx))$”, and in this the outer function $F$ and the inner function $F$ must have different meanings; for the inner has the form $ϕ(fx)$, the outer the form $ψ(ϕ(fx))$. Common to both functions is only the letter “$F$”, which by itself signifies nothing.

This means that Russell's paradox, amounting at : $R(R) \text { iff } \lnot R(R)$ is a violation of the syntactical rules of the formalized language, because the rules prohibit that a function of "level" $n$ can be an argument of a function of the same "level".

See also 3.332 :

No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the “whole theory of types”).