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Consider the following sets:

S1 = {S | S is a set such that S ∉ S}

S2 = {S | S is a set}

Do S1 & S2 exist?

The answer is both do not exist. I do not understand why so? S ∉ S is always true as {A} ∉ {A} so S1 can have elements and sets can exist as elements in a set, so why does S2 doesn’t exists either?

S1 = {S | S is a set such that S ∉ S}

S2 = {S | S is a set}

Do S1 & S2 exist?

The answer is both do not exist. I do not understand why so? S ∉ S is always true as {A} ∉ {A} so S1 can have elements and sets can exist as elements in a set, so why does S2 doesn’t exists either?

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Best answer

Bigger question is whether S1 is in S1 or not? That’s a classical paradox.

You can read more about it here : **Russell’s paradox**.

That’s why such S1 does not exists.

Same way, S2 is a never ending loop, as S2 is a set, then S2 must contain itself, the new S2 must contain itself.. this goes on forever and you can never contain S2 inside S2.

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