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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?

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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
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1 Answer

up vote 23 down vote accepted

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:

  1. ∅ is a subset of every set
  2. 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?

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