Bertrand Russell's Set of Everything
Bizarrely, Bertrand Russell once set out to calculate how many things there are in the universe. In order to do so, he was lead to conclude that there must be a set which literally includes everything!
In actual fact, Russell reasoned in this rather bizarre way in order to counteract George Cantor's view that there is no largest number. In response, Russell claimed that if he had discovered (as it were) a set which included everything, and the nature of sets determine the nature of numbers (or sets literally are numbers), then if that set included everything, then that set is the largest number!
Clearly there is a strange notion here that the members of a set somehow determine that set and determine numbers and therefore, in this case, the largest number.
Did Russell also mean by “everything” every concrete or empirical thing? If he had included abstract objects or entities in that set, then surely it could not have included everything because numbers, for one, are infinite... but that's begging the question! According to Russell, and Frege before him, numbers are sets. Therefore what makes up a set determines what makes a number. And if everything is included in a set, and that everything is finite, then it also determines the finitely largest number.
You cannot call in abstract numbers here because it is the nature of sets that determines the nature and existence of numbers. More simply, that Set of Everything cannot have numbers as members because the sets themselves are numbers. (However, what about other abstract entities such as universals, properties, possible worlds, etc.?)
The Set of All Sets Which Don't Include Themselves
This Russellian insight, against Cantor, also led to a paradox which worked against the Fregean notion of sets as well as his attempt to reduce the whole of mathematics to logic.
Firstly, one we have the Set of Everything, an obvious question to ask is this: if the Set of Everything includes everything, then it must also include sets. Now the Set of Everything is, well, a set: therefore must it include itself? In order for the Set of Everything to literally include everything, it must include itself as a member because it is part of that everything.
Now this also throws up another problem.
If the Set of Everything includes sets, including itself as a set, then does it also include the sets which don't include themselves? Remember we are talking about the Set of Everything here. Therefore it must include both sets which are members of themselves and sets which aren't members of themselves. We've already said that because the Set of Everything includes everything then it must include itself. It thus becomes a set which includes itself. But, because it must, again, include everything, it must also include sets which don't include themselves. That means that it must include a set which doesn't include itself. But is the Set of Everything such a set? It can't be, because it includes everything – including itself. There could be a set which doesn't include itself outside the Set of Everything; though because the Set of Everything includes everything, we have already established that this is impossible.
Thus if the Set of Everything contains itself as a member, then it doesn’t include itself as a member. This means that the Set of Everything cannot exist! If it contains itself, it doesn't contain a set, itself, which is not a member of itself. But if it is not a member of itself, then it is a member of itself because it contains all sets.