TITLE: What is the cardinality of the set of the empty set?
QUESTION [8 upvotes]: Given set $ A = \{  \emptyset   \}$ what is $|A|$? I just learned about sets today in class and I'm unsure how to answer this.

REPLY [3 votes]: In a way it is the start of the construction of natural numbers (cardinalities of finite sets)  


  0=|
∅
  |,
1=|{
∅
}|,
2=|{
∅
  ,{
∅
  } }|,
3=|{
∅
  ,{
∅
  },{
∅
  ,{
∅
  } }  }|,...
 

as you see the first set (between |.|) has no elements, the second one has one element, the third one has two distinct elements as the empty set and the set whose only element is an empty set are different. The (usually confusing) ... just adds the nested braces.