In naive set theory ∅ = {∅} = {{∅}}? In naive set theory, I believe ∅ = {∅} = {{∅}} is correct, but just wanted to make sure that I understood this correctly. 
∅ is an empty set, so having an empty set as an element of a set that contains nothing else is pretty much the same thing as e.g. marking the number 2 as 2.000000, right? Doesn't the symbol ∅ intrinsically imply that this is an empty set which contains it self?
 A: No, it is incorrect.


*

*∅ is the empty set.

*{∅} is a set, containing exactly one item: The empty set.

*{{∅}} is a set, containing exactly one item: A set with one item, which is the empty set.



Doesn't the symbol ∅ intrinsically imply that this is an empty set which contains it self?

You're confusing two things here: set membership and subsets:


*

*∅ is a subset of every set

*but it is not a member of every set, just like 1 is not a member of every set either


Example
If you have two items, a and b, and you are to construct the set of all possible combinations, choosing 0 to all items, this will be your solution:

{∅, {a}, {b}, {a, b}}

Naturally, every possible combination is represented by a set, that contains the chosen items. And the set of all possible combinations is (obviously) represented by a set containing all those combinations (i.e. sets), now we have a set of sets.


*

*We can choose no item at all: ∅ is part of our solution

*We can choose one item: {a} and {b} are part of the solution

*We can choose both items: {a, b}
Note: This is called the power set of {a, b}, usually denoted P({a, b}).
Maybe you think of ∅ as "nothing", because it's empty. However, that's quite far from the truth, an empty set is very "real", it's not nothing. You wouldn't say an empty glass is nothing, would you?
A: No, it's not the same
• ∅ represents the empty set.
• {∅} Is a non-empty set. It contains one element. That element is the empty set
• {{∅}} This set contains one element. Such element is the set containing the empty set.
Another analogy that might me useful is to think of set as bags. {∅} would represent an empty bag and {{∅}} would represent a bag containing an empty bag
