In naive set theory ∅ = {∅} = {{∅}}?

Question

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?

Somewhat related: this answer, this answer and this question. — Martin Sleziak, Sep 26 '12 at 12:19
It may help to think of a set as being like a piece of paper with stuff written on it. ∅ is a blank sheet of paper. {∅} is a sheet of paper that has "∅" written on it. {{∅}} is a sheet of paper that has "{∅}" written on it. — Tanner Swett, Sep 27 '12 at 21:05

score 33 · Accepted Answer · answered Sep 26 '12 at 08:14

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?

score 0 · Answer 2 · answered Aug 26 '20 at 11:25

0

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

answered Aug 26 '20 at 11:25

user926356

1,250

Your bag analogy is one bag short. $\emptyset $ being the empty set is already one bag, so ${\emptyset}$ is a bag containing an empty bag, and so on. – user400188 Mar 07 '21 at 23:19

In naive set theory ∅ = {∅} = {{∅}}?

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