This is a variation on Russel's (an actually good English philosopher, which is almost a paradox by itself 
) quite acute comment on the set theory used by Cantor, and (to not sound more pompous than absolutely needed) it states that
"if a set exists for all sets that are not part of themselves (elaboration on that in a bit) then that set can neither be part of itself or not part of itself".
A brief elaboration on that follows:
In Cantor's set theory the sets are either of type X, ie include elements which do not also have the set itself as part of them, or of type Y, ie include elements as well as the actual set including them.
You may ask what a set including its own self can be. A rather direct example i read is of a set that includes all squares which can be formed in a plane. Those squares (unless limited by the definition) are infinite, cause a plane can have countless squares of whatever perimeter. Such a set is not including its own self there, though, cause the set of those squares is obviously something else and not another square on that plane.
By contrast if we name a set as "the set of everything that is not a square on that plane", then the set does indeed also include its own self, for as an object it is not a square on that plane. But so is any other arbitrary thing, eg a human, or a flower, or any triangle on that plane, or any triangle not on that plane, or any square not on that plane, and so on.
Russel noted that it is ambiguous and incomplete to have a notion of a set that collects anything being left out of all other sets in a theory, and thus the theory of sets would not be true to its own system of definition of sets.
In essence the question is one of the vast difference between something "being" something, and something "not being" something, cause in the former case something can be seen as a type or nearing such a type, whereas in the latter case there is (crucially, non trivially) no larger degree of difference in non-being X regardless if you are Y, Z or any infinite variation in between.
What follows is a more descriptive analogous formation of Russel's paradox, titled as the paradox of the complete catalogue of catalogues. Useful to note it was worded in 1908, so a couple of decades prior to Borges' famous short story about a library with infinite volumes of books.
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You can comment on the issues presented here, either on set theory or language and the hugely crucial difference between being and non-being, which was a central issue in Eleatic philosophy too, with Parmenides having as one of his main axioms that "Being exists, and non-being is nothing at all", hence protecting himself from the cutting edges of a similar paradox in his view of Oneness. Likewise in set theory without restrictive (non purely logic following) axioms we cannot have the totality of sets.
) quite acute comment on the set theory used by Cantor, and (to not sound more pompous than absolutely needed) it states that "if a set exists for all sets that are not part of themselves (elaboration on that in a bit) then that set can neither be part of itself or not part of itself".
A brief elaboration on that follows:
In Cantor's set theory the sets are either of type X, ie include elements which do not also have the set itself as part of them, or of type Y, ie include elements as well as the actual set including them.
You may ask what a set including its own self can be. A rather direct example i read is of a set that includes all squares which can be formed in a plane. Those squares (unless limited by the definition) are infinite, cause a plane can have countless squares of whatever perimeter. Such a set is not including its own self there, though, cause the set of those squares is obviously something else and not another square on that plane.
By contrast if we name a set as "the set of everything that is not a square on that plane", then the set does indeed also include its own self, for as an object it is not a square on that plane. But so is any other arbitrary thing, eg a human, or a flower, or any triangle on that plane, or any triangle not on that plane, or any square not on that plane, and so on.
Russel noted that it is ambiguous and incomplete to have a notion of a set that collects anything being left out of all other sets in a theory, and thus the theory of sets would not be true to its own system of definition of sets.
In essence the question is one of the vast difference between something "being" something, and something "not being" something, cause in the former case something can be seen as a type or nearing such a type, whereas in the latter case there is (crucially, non trivially) no larger degree of difference in non-being X regardless if you are Y, Z or any infinite variation in between.
What follows is a more descriptive analogous formation of Russel's paradox, titled as the paradox of the complete catalogue of catalogues. Useful to note it was worded in 1908, so a couple of decades prior to Borges' famous short story about a library with infinite volumes of books.
Paradox of the complete catalogue of catalogues said:
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You can comment on the issues presented here, either on set theory or language and the hugely crucial difference between being and non-being, which was a central issue in Eleatic philosophy too, with Parmenides having as one of his main axioms that "Being exists, and non-being is nothing at all", hence protecting himself from the cutting edges of a similar paradox in his view of Oneness. Likewise in set theory without restrictive (non purely logic following) axioms we cannot have the totality of sets.
