In naive set theory ∅ = {∅} = {{∅}}?
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?