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.