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While I'm reading Is an empty set equal to another empty set?, I wonder if {∅} = ∅?

And it leads to my doubts : is there such things as "set of empty set" with notation {∅}?

If so, then {∅} = ∅?

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Think of a box. Empty set is like an empty box. But a box with another box inside it is no longer empty.

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  • $\begingroup$ I used to understand empty set as "nothing". So "Nothing of nothing" is still "nothing". That was my thinking. While understanding empty set as empty box everything makes sense. So "empty set" shouldn't be "nothing" at all? $\endgroup$ – user2829759 Oct 2 '15 at 8:23
  • $\begingroup$ That's right, it shouldn't. It's a box with nothing inside it. $\endgroup$ – Ivan Neretin Oct 2 '15 at 8:36
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$\{\emptyset \}$ has one element, while $\emptyset$ has $0$ elements.

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In your language, the empty set can not be "nothing at all" since it is really a mathematical object: a set with no elements, as you understood thanks to the box example. The fact it does not contain anything does not affect the fact it is a set.

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