How can it be that the empty set is a subset of every set but not an element of every set? How can it be that the empty set is a subset of every set but not an element of every set?
I've understood that the empty set must be a subset of every set because if it were not a subset of every set then the statement $\exists x : x \in \emptyset \land x \notin M$, would need to be ture, but since the empty set has no elements this statement is a contradiction.
But how can it be that the empty set is not element of every set, if it is a subset of every set?
I also understood that it cannot be an element of itself, otherwise it wouldn't be the empty set anymore, but why is it not an element of every non-empty set?
If the empty set is subset of every set, than it also would need to be a subset of itself ($\emptyset \subset \emptyset$), how can that be?
 A: *

*There might be versions of set theory where the requirement " the empty set is an element of every set" is satisfied. What I mean is that  it does not seem absurd prima facie. For example, in the set theoretic consruction of natural numbers, number zero ( that is : the empty set) is an element of every ( natural) number greater than 0 , and these numbers are sets. ( for example , $1=\{\emptyset\}= \{0\}, 2= \{\emptyset, \{\emptyset\}\}=\{0,1\} , 3= \{0,1,2\} $.


*However, the question " is every set a member of every set ( different from itself)?"  can be settled  as a pure matter of fact. Any counter-example would do; Consider, for example,  the set : $\{1, 2,3\}$.


*I think the question is : why does it seem plausible that, if a set is a subset of every set, then it should also be an element of every set? Maybe one could try to reconstruct the reasoning that produces this false appearence :

(1) The empty set s a subset of every set, say, of set S
(2) Therefore, all the elements of $\emptyset$ are also elements of S.
(3) Therefore the totality of the elements of $\emptyset$ is an
element of S.
(4) But this totality is $\emptyset$  itself .
(5) Therefore $\emptyset\in S$.


*

*The mistake is hidden in steps  (3) and (4).

a) as to (3) : though it is true distributively that every element $\emptyset$ is an element of S, it is not true collectively
b) as to (4) the totality of the elements of $\emptyset$ is nothingness ( nil, nothing) , and therefore is not the same thing as the empty set which is something ( namely, a set).
A: This is a crude analogy but I think it gives a concrete explanation of the idea.
Think about a set as a bowl full of things.
A subset is what you get when you take some things out of the bowl.
Take everything out of the bowl, and the bowl is empty. This corresponds to the empty set.
Every bowl can be emptied. So the empty set is a subset of every set.
A = {}
Set A is the empty set. Since it's empty, it can't contain the empty set.
B = { 1, {} }
Set B actually contains the empty set.
C = { {1}, {1, {}} }
Set C can also be written as C = {{1}, B}. Set B contains the empty set but set C does not.
