# Sets and well-defined objects

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.

• 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. Oct 3, 2021 at 14:26
• 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.) Oct 3, 2021 at 14:31
• 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? Oct 3, 2021 at 14:34
• Oct 19, 2021 at 9:15

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