Identity.

Wittgenstein's notation does not use a sign of identity, but expresses the identity of an object by using the identical sign for it; if the objects are different, so are the signs. Identity is not a relation between objects. This becomes very clear if one considers the example: "All x such that x satifies the function f and x=a."

((x):fx.⊃.x = a).

This proposition simply says: "Only 'a' satisfies the function 'f', and not that only things that have a certain relation to 'a' satisfy the function. Of course, it might then be said that only 'a' has this relation to 'a'; but in order to express that, we would have to use the identity sign itself. Russell's definition of '=' is inadequate for this purpose, because according to it we cannot say that two objects have all their properties in common. (Even though this proposition is never correct, it still makes sense.)

By the way: to say two things are identical is absurd, and to say that one thing is identical with itself is to say nothing at all.

In Wittgenstein's notation, one does not write (f(a, b).a = b) but (f(a, a)) or (f(b, b)).
Not (f(a,b).~a = b) but (f(a, b)).
Analogously, one does not write ((∃x, y). f(x, y). x = y), but ((∃x).f(x, x));
and not ((∃x, y).f(x, y)).~x = y), but ((∃x, y).f(x, y)).

So instead of Russels ((∃x,y).f(x,y)) one writes ((∃x,y).f(x,y).∨.(∃x).f(x,x)).
Instead of ((x):fx⊃x = a) one writes ((∃x).fx.⊃:(∃x, y).fx.fy).
And the proposition, 'Only one x satisfies f( )', will read ((∃x).fx:~(∃x, y).fx.fy).

The identity-sign, therefore, So a sign of identity is not essential in a correct conceptual notation, and pseudo-propositions like (a=a), (a=b.b=c.⊃a=c), (x).x = x), ((∃x).x = a), etc. cannot even be written down. This also disposes of all the problems that were connected with them.

All the problems with Russell's 'axiom of infinity' can be solved now by considering that it would be expressed in language by the existence of infinitely many names with different meanings.