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Long back, I learned the first definition of a set as "A Set is a collection of well-defined objects". If I go with this definition, then I want to understand basic reason in argument.

Let $S$ be the collection of all sets $A$ such that $A$ do not contain itself.

It is known that $S$ can not be a set. My question is very basic one; what should be proper justification for it (among below)?

i) $S$ can not be a set because $S$ is not collection of well-defined objects.

ii) $S$ is a collection of well-defined objects, but if it is a set, then $S\in S$ implies $S\notin S$, and vice-versa, hence, it is not a set.

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    $\begingroup$ Notice that if you go with your stated definition, then ii) can be restated as "$S$ is a set, but if it is a set...", which doesn't make sense, so the issue here is that the elements of $S$ are not well-defined. $\endgroup$
    – DMcMor
    Oct 3, 2021 at 14:26
  • $\begingroup$ Thank you very much for valuable comments. (I usually see books, and after contradiction, it is mentioned that $S$ is not a set - in Russel's paradox. But, in step-by-step arguments, if we are getting some contradiction, it is natural to ask from which step contradiction is arising; so I posted this question. Thanks again for your comments.) $\endgroup$ Oct 3, 2021 at 14:31
  • $\begingroup$ But, a comment: if we take a set $A:=\{2\}$, then the set $A$ is not a member of itself; then how can we say that the objects in definition of $S$ in question are not well-defined? $\endgroup$ Oct 3, 2021 at 14:34
  • $\begingroup$ See Self-studying Russell's Paradox. $\endgroup$
    – user21820
    Oct 19, 2021 at 9:15

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The problem is the "definition" of a set. In modern set theory one does not define what a set is, but requires that various axioms are satisfied. Make a search for "axiomatic set theory" here in math.stackexchange. What you try to do is to use the axiom schema of specification. But this only applies to form subsets of a given set; collections of objects having a certain property are in general no sets.

Let S be the collection of all sets $A$ such that $A$ do not contain itself.

These objects are well-defined. But the resulting "collection" is not a set as your argument in ii) shows. In some axiomatic approaches such a general collection as here is called a class. Intuitively, some classes may be too big to have the property of being a set.

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