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    Prove that "A set of all sets does not exist "

    I know its better to use proof by contradiction here. So I assume a universal set which contains all the sets exist .However after that I don't know what to do ?? Can someone point me in the right direction?
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    I know very little about sets, but I don't think a set can have itself as a member. I'm probably wrong, so please say so (unless I'm right of course).
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    (Original post by meatball893)
    I know very little about sets, but I don't think a set can have itself as a member. I'm probably wrong, so please say so (unless I'm right of course).
    I think what you said is right ,For instance let the universal set be A , since A doesn't belong to A itself it should belong to some other unknown set. Hence a universal set does not exist ..But i dont know if its a valid proof.
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    If there was a set that contained all the sets then it must contain itself because it itself, is a set. To not contain itself would falsify the idea that it is a set of all sets. The question is really "Can a set contain itself?" I cant remember how you prove it though.
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    Wiki russell paradox or you could consider the power set.

    Yeah, if you have a set X, where X=m and contains all sets. But, the power set P(x) would be bigger then set X, from |P(x)|=2^m. So P(x) should be bigger then set X which is impossible as set X should contain P(x) because it is the set of all sets. Contradiction.
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    Recall the naive comprehension axiom: for any well-defined predicate P(x), \exists a \forall x ( x \in a \leftrightarrow P(x) ). Define the predicate P(x) such that P(x) := x \notin x, and let the set r be a set which must exist by the NCA. Now, we obviously have that either r \in r or r \notin r. Your task is to show that both of these cases lead us to a contradiction.
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    ahhaah i remember this. my lecturer was giving us a brief history of mathematics and he told us about this. we just didn't care :P
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    So what about the universal set?

    In set theory, a universal set is a set which contains all objects, including itself.
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    From the same article:
    Zermelo–Fraenkel set theory and related set theories, which are based on the idea of the cumulative hierarchy, do not allow for the existence of a universal set.
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    (Original post by nuodai)
    So what about the universal set?
    The OP is almost certainly working with first-order logic.
 
 
 

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