### Logic says...

Say, we found the "set of all sets that do not contain themselves" and we name it $A$.

Let us try to list out the elements of this set.
$$
A = \{A_1, A_2, ... , A_{\infty}\}
$$
But something is missing. Ask yourself, "What is $A$? What are its properties?"

You can say many points about $A$, one of them being:

$A$ does not contain itself.

This is right. That means "$A$ is a set that doesn't contain itself"

What does this also mean? It means that $A$ is **not** a set that contains "**all** sets that do not contain themselves" (because it doesn't contain $A$, that also satisfies the property).

Hence, $A$ is not the set that we were looking for.

Let us consider another set $B$, that contains $A$ as an element, and the elements of $A$ too. So, it looks something like this

$$
B = \{A, A_1, A_2, ... , A_{\infty}\}
$$
But again, looking at $B$, you will find that there can be a set
$$
\{B,A, A_1, A_2, ... , A_{\infty}\}
$$
that contains sets that do not contain themselves.

Thus, we can always find one such set, and make it a member of another such set. We can keep going iteratively without any bounds or termination.

So, we cannot end up with an answer. Because there always exists a better answer for correcting the current answer's **one minor flaw**.

Again, this is a paradox, so we can't **get** an answer, neither we can prove nor disprove whether the answer exists.