Intersection set theory: Encyclopedia II - Intersection set theory - Nullary intersection

Intersection set theory - Nullary intersection

Note that in the previous section we excluded the case where M was the empty set (∅). The reason is the follows. The intersection of the collection M is defined as the set (see set-builder notation)

If M is empty there are no sets A in M, so the question becomes "which x's satisfy the stated condition?" The answer seems to be every possible x. When M is empty the condition given above is an example of a vacuous truth. So the intersection of the empty family should be the "set of everything". The problem is, there is no such set. Assuming such a set exists leads to a famous problem in naive set theory known as Cantor's paradox. For this reason the intersection of the empty set is left undefined. There is nothing that can be done about the problem, it is just a fact of life in mathematics.

A partial fix for this problem can be found if we agree to restriction our attention to subsets of a fixed set U called the universe. In this case the intersection of a family of subsets of U can be defined as

Now if M is empty there is no problem. The intersection is just the entire universe U, which is a well-defined set by assumption.

Adapted from the Wikipedia article "Nullary intersection", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki