Bertrand Russell, answered from the texts and cited to the page.
The paradox emerged from an attempt to take Cantor's diagonal argument seriously. In 1901, working toward the derivation of all mathematics from purely logical foundations, I applied Cantor's proof that there is no greatest cardinal number to what I supposed was the class of all objects whatsoever.1 That class, if it existed, would have to embrace itself as one of its members — since "everything" is something, and must therefore belong to the class of everything.2
From this I was led to a simpler and more direct contradiction. Consider that normally a class is not a member of itself: mankind is not a man.2 Now form the class of all classes which are not members of themselves. Ask whether this class is a member of itself or not. If it is a member of itself, then it is one of those classes that are not members of themselves — so it is not.
If it is not a member of itself, then it is not one of those classes that are not members of themselves — so it is.3 Each hypothesis implies its contradictory, and there we are stuck. The same contradiction can be stated in terms of predicates rather than classes. Take the predicate "not predicable of itself." If that predicate is predicable of itself, it is not; if it is not, it is.4
The barber version — defined as the one who shaves all and only those who do not shave themselves — is sometimes offered as equivalent, but it is easier to dissolve: one simply denies that any such barber exists.5 The class version is not so easily dismissed. The resolution I eventually reached is that the question whether a class is or is not a member of itself is not false but meaningless — pure noise without significance.5
A class is an incomplete symbol, in the same sense as a definite description: in any fully analysed proposition that appears to be about a class, the class itself does not appear; it has been broken up and dissolved into something else.5 The expression "this class is a member of itself" cannot be translated into that fully analysed form, which is precisely why it generates contradiction rather than falsehood.
The theory of types follows from this. Objects must be arranged in a hierarchy — individuals, classes of individuals, classes of classes, and so on — and a propositional function is only meaningful when its argument belongs to the appropriate level of that hierarchy.6 The expression x ∈ x is meaningless because membership requires that the class be of a higher type than its members; you cannot coherently ask whether a thing belongs to itself any more than you can ask whether a colour weighs more than a tune.6
The theory of types, I should add, is not a finished structure: much of it remains inchoate, and the precise form it should take is less certain than the need for something of the kind.7
When I first came upon this contradiction, in the year 1901, I attempted to discover some flaw in Cantor's proof that there is no greatest cardinal.The Collected Papers Vol 4, Foundations of Logic 1903-05, pp. 7–8
The comprehensive class we are considering, which is to embrace everything, must embrace itself as one of its members. In other words, if there is such a thing as 'everything,' then 'everything' is something, and is a member of the class 'everything.' But normally a class is not a member of itself. Mankind, for example, is not a man.Introduction to Mathematical Philosophy
If it is, it is one of those classes that are not members of themselves, i.e. it is not a member of itself. If it is not, it is not one of those classes that are not members of themselves, i.e. it is a member of itself. Thus of the two hypotheses—that it is, and that it is not, a member of itself—each implies its contradictory.Introduction to Mathematical Philosophy
If x be a predicate, x may or may not be predicable of itself. Let us assume that 'not-predicable of oneself' is a predicate. Then to suppose either that this predicate is, or that it is not, predicable of itself, is self-contradictory.The Principles of Mathematics
You can define the barber as 'one who shaves all those, and those only, who do not shave themselves'... But in our previous form I think it is clear that you can only get around it by observing that the whole question whether a class is or is not a member of itself is nonsense, i.e. that no class either is or is not a member of itself, and that it is not even true to say that, because the whole form of words is just a noise without meaning.Part II, p. 274
it is necessary to distinguish various types of objects, namely terms, classes of terms, classes of classes, classes of couples of terms, and so on; and that a propositional function φx in general requires, if it is to have any meaning, that x should belong to some one type. Thus xex was held to be meaningless, because e requires that the relatum should be a class composed of objects which are of the type of the referent.The Principles of Mathematics
Now the theory of types emphatically does not belong to the finished and certain part of our subject: much of this theory is still inchoate, confused, and obscure. But the need of some doctrine of types is less doubtful than the precise form the doctrine should take.Introduction to Mathematical Philosophy