Subject: Empty Sets
From: "William J. Rapaport" <rapaport@buffalo.edu>
Date: Fri, 13 Nov 2009 19:50:53 -0500 (EST)

A student writes:
"I just wanted to be sure that "Every empty set contains an empty set,
and therefore has exactly 1 element." is a true statement. So, if I
were to say 'An empty set contains an empty set, which contains an
empty set, and so on..', this would be correct?"

Reply:  
It would depend on what you mean by "contains". "Contains" could mean:
contains as an element, or: contains as a subset. However, on both
interpretations, what you said is not quite correct.

Every empty set contains as elements nothing; put a bit more
idiomatically: An empty set has no elements, no members. So, it has 0
elements. Moreover, it does not contain anything, so, in particular, it
does not "contain an empty set".

On the other hand, an empty set has exactly one subset, namely, itself,
as we discussed in lecture today. And, indeed, an empty set contains as
subset an empty set, which contains as subset an empty set, etc.

By comparison, consider the set {1}. Its power set is:  { {}, {1} }.
And the power set of that 2-element set is:
{ {}, {{}}, {1}, {{},{1} }, etc. 
The sets of subsets of subsets...  gets bigger and bigger.
Not so for the empty set.