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I think I understand the way in which Russell's paradox shows that the following principle is wrong:
" for every predicate, there is a set having as elements all the objects that satisfy this predicate"
Russell's picks up a predicate ( namely the predicate " x is not an element of itself" ) and shows that the corresponding " set" would have contradictory properties, which means that " the set of all x such that x is not an element of x" does not exist.
This counts as a counter example to the alledged " principle".
My understanding of Russell's paradox does not go further than this.
Now, if I am correct, it is often said that Russell showed, with this paradox, that the " set of all sets" does not exist.
What is actually the relation between Russell's paradox and the non-existence of the set of all sets?
It seems difficult to me to answer that the relation consists in the fact that precisely a set has the property of not belonging to itself. For it seems to me that Russell's paradox forbids to define a set in this way.